Effective elastic modulus of wet granular materials derived from modified effective medium approximation and proposal of an equation for the friction coefficient between the object and wet granular materials surfaces

Abstract

Liefferink et al. (GM 22: 57, 2020) reported the effect of liquid volume fraction on the penetration hardness/shear modulus and the friction coefficient of several types of wet sand. However, no analytical equation can simultaneously explain these two observations. In this paper, we formulate the effective elastic modulus of wet granular material as a function of the liquid volume fraction using the effective medium approximation. We also propose the equation of the friction coefficient between the surface of an object and the surface of the wet granular materials. The equations for the modulus of elasticity and coefficient of friction as a function of the liquid volume fraction can well explain the observed data by using a few adjusted parameters.

Graphic abstract

Appendix

We will show approximately that Eq. (10) can be derived from Eq. (1) with the assumption that the effective Poisson’s ratio can be considered not to change significantly. With this assumption, Eqs. (1) and (2), which are coupled equations become two separate equations. Equation (1) can be written in the following form

$$\frac{{p_{dd} G_{e} }}{{G_{e} /\beta_{e} + \left( {G_{dd} - G_{e} } \right)}} + \frac{{p_{dw} G_{e} }}{{G_{e} /\beta_{e} + \left( {G_{dw} - G_{e} } \right)}} + \frac{{p_{ww} G_{e} }}{{G_{e} /\beta_{e} + \left( {G_{ww} - G_{e} } \right)}} = \beta_{e}$$

(32)

With several mathematical steps, Eq. (32) can be rewritten as

$$\begin{gathered} p_{dd} \frac{{G_{dd} - G_{e} }}{{G_{dd} + \left( {1/\beta_{e} - 1} \right)G_{e} }} + p_{dw} \frac{{G_{dw} - G_{e} }}{{G_{dw} + \left( {1/\beta_{e} - 1} \right)G_{e} }} \hfill \\ + p_{ww} \frac{{G_{ww} - G_{e} }}{{G_{ww} + \left( {1/\beta_{e} - 1} \right)G_{e} }} = - \beta_{e} + \frac{{p_{dd} }}{{1 + \left( {1/\beta_{e} - 1} \right)G_{e} /G_{dd} }} \hfill \\ + \frac{{p_{dw} }}{{1 + \left( {1/\beta_{e} - 1} \right)G_{e} /G_{dw} }} + \frac{{p_{ww} }}{{1 + \left( {1/\beta_{e} - 1} \right)G_{e} /G_{ww} }} \hfill \\ \end{gathered}$$

(33)

Let us look at the right side of Eq. (33). If the volume of the liquid fraction is close to zero (the granular system is almost dry), $p_{dd} \approx 1$, $p_{dw} \approx p_{ww} \approx 0$, and $G_{e} = G_{dd}$. As a result, the right side of Eq. (33) becomes $\approx 0$. If the volume fraction of the liquid is very high so that $p_{ww} \approx 1$, $p_{dd} \approx p_{dw} \approx 0$, and $G_{e} = G_{ww}$ so that the right-hand side of Eq. (33) is also close to zero. For the case of wet granular materials, $G_{dw} > G_{dd}$ and $G_{dw} > G_{ww}$. When the shear modulus approaches the peak, $p_{dw}$ is not too small and $G_{e}$ is close $G_{dw}$ but smaller. If this condition is reached and we approximate $G_{e} /G_{dd} \to \infty$ and $G_{e} /G_{dd} \to \infty$ (this is actually a rough approximation), the right-hand side of Eq. (33) approaches

$$- \beta_{e} + \frac{{p_{dw} }}{{1 + \left( {1/\beta_{e} - 1} \right)G_{e} /G_{dw} }}$$

Since $p_{dw} = O\left( 1 \right)$, $1 + \left( {1/\beta_{e} - 1} \right)G_{e} /G_{dw} < 1 + \left( {1/\beta_{e} - 1} \right) = 1/\beta_{e}$, we obtain $p_{dw} /\left[ {1 + \left( {1/\beta_{e} - 1} \right)G_{e} /G_{dw} } \right] \approx \beta_{e} O\left( 1 \right)$ and the right-hand side of Eq. (33) is approximately $\approx - \beta_{e} + \beta_{e} O\left( 1 \right) \approx 0$. From this analysis we can conclude that Eq. (33) can be approximated by the following equation

$$p_{dd} \frac{{G_{dd} - G_{e} }}{{G_{dd} + \left( {1/\beta_{e} - 1} \right)G_{e} }} + p_{dw} \frac{{G_{dw} - G_{e} }}{{G_{dw} + \left( {1/\beta_{e} - 1} \right)G_{e} }} + p_{ww} \frac{{G_{ww} - G_{e} }}{{G_{ww} + \left( {1/\beta_{e} - 1} \right)G_{e} }} \approx 0$$

(34)

Each term of Eq. (34) can be written as

$$p_{i} \frac{{G_{i} - G_{e} }}{{G_{i} + \left( {1/\beta_{e} - 1} \right)G_{e} }} = p_{i} \frac{{\frac{{\left( {1 + v_{e} } \right)}}{{\left( {1 + v_{i} } \right)}}2\left( {1 + v_{i} } \right)G_{i} - 2\left( {1 + v_{e} } \right)G_{e} }}{{\frac{{\left( {1 + v_{e} } \right)}}{{\left( {1 + v_{i} } \right)}}2\left( {1 + v_{i} } \right)G_{i} + \left( {1/\beta_{e} - 1} \right)2\left( {1 + v_{e} } \right)G_{e} }}$$

(35)

The Poisson ratios of most sand granules are approximately 0.3–0.4 [31]. Assuming that the effective Poisson’s ratio does not differ much from this value, we can approximate $\left( {1 + v_{e} } \right)/\left( {1 + v_{i} } \right) \approx 1$ and we get the equation

$$\mathop \sum \limits_{i = 1}^{3} p_{i} \frac{{E_{i} - E_{eff} }}{{E_{i} + \left( {1/\beta_{e} - 1} \right)E_{eff} }} \approx 0$$

(36)

where $i = dd$, $dw$, and $ww$. This approximate equation is very similar to the EMA equation which has been used to explain the formation of electrical conductivity in composites [28,29,30],

$$\mathop \sum \limits_{i = 1}^{3} p_{i} \frac{{E_{i} - E_{eff} }}{{E_{i} + \left( {z/2 - 1} \right)E_{eff} }} = 0$$

(37)

by analogizing $1/\beta_{e}$ as $z/2$, where $z$ is the coordination number. This analogy does not produce very different estimates.

As mentioned above, the value of $\beta_{e}$ for sand granules is in the range of 0.44–0.48. The Poisson’s ratio for most materials varies from zero (for very brittle materials) to 0.7 (a completely elastic material) [46] so, assuming $v_{e}$ is in that range, then the range of values for $\beta_{e}$ is between 0.22 and 0.53. Taking $z \approx 4.5 - 7.5$ [40, 41], we obtain $2/z$ between 0.27 and 0.44, which is not much different from $\beta_{e}$. According to Hill [20], $\beta_{e}$ is in the range 0.4–0.6; is very near 0.45 for most metals $\beta_{e}$. An interesting result is the report of Spinner et al. [47] on the effect of porosity on the Poisson’s ratio of porous thorium oxide (ThO₂), where the Poisson’s ratio decreases with increasing porosity. Hill [20] has obtained the dependent equation of Poisson’s ratio to volume fraction of one component of the composite composed of two components according to the equation $v_{e} = \left( {p_{1} v_{1} + p_{2} v_{2} - v_{1} v_{2} } \right)/\left( {1 - p_{1} v_{2} - p_{2} v_{1} } \right)$. For porous materials, the second component is a vacuum so we can take $v_{2} = 0$. If porosity is $\phi$, $p = 1 - \phi$ so $v_{e} = \left( {1 - \phi } \right)v_{1} /\left( {1 - \phi v_{1} } \right)$. It is easy to prove that $v_{e}$ decreases with increasing $\phi$ in the range $0 \le \phi \le 1$. Martin et al. [48] also proved that for porous ZnO, the Poisson’s ratio decreases with increasing porosity according to the empirical equation $v = v_{0} \left( {1 - \phi } \right)^{\tau }$, where $v_{0}$ is the porosity of the solid material and $\tau = 1.12$. These results can be understood because the greater the porosity, the more brittle the material is so that the Poisson’s ratio is getting smaller. We can easily show that $\beta_{e}$ gets bigger as Poisson ratio gets smaller. And intuitively, because the volume of the material is getting smaller when the porosity is greater, the greater the porosity, the lesser dense the material which causes the coordination number to be smaller. So, we conclude that the greater the porosity of the material, the smaller $1/\beta_{e}$ according to the change in $z/2$, which supports the analogy of $1/\beta_{e}$ as $z/2$ is reasonable.

Indeed, the value of $z/2$ is not exactly equal to $1/\beta_{e}$. For equations that have been derived by a number of approximations, these differences could be tolerated. By looking at the similarities above, we will use Eq. (10) to estimate the elastic modulus of the granular material. Thus, the equation for estimating the modulus of elasticity proposed here is different from that reported by many researchers [12, 13, 20, 49,50,51,52,53,54].