Micro and macro mechanical characterization of artificial cemented granular materials

Abstract

The focus of this study is the experimental characterization of cemented granular materials, with the aim of identifying the microscopic properties of the solid bonds and describing the extension to macroscopic mechanical strength of cemented samples. We chose to use artificially bonded granular materials, made of glass beads connected by solid paraffin bridges. The results of several sets of laboratory tests at different scales are presented and discussed. Micromechanical tests investigate the yield strength of single solid bonds between particles under traction, shearing, bending and torsion loading, as a function of variations in particle size, surface texture and binder content. Macro-scale tensile tests on cemented samples explore then the scale transition, including influence of confining walls through homothetic variations of the sample size. Despite the large statistical dispersion of the results, it was possible to derive and validate experimentally an analytical expression for micro tensile yield force as a function of the binder content, coordination number and grain diameter. In view of the data, an adhesive bond strength at the contact between bead and solid bond is deduced with very good accuracy and it is even reasonable to assume that the other threshold values (shear force, bending and torsion moments) are simply proportional to the tensile yield, thus providing a comprehensive 3D model of cemented bond. However, the considerable dispersion of the data at the sample scale prevents validation of the extended model for macroscopic yield stress. A final discussion examines the various factors that may explain intrinsic variability. By comparison with other more realistic systems studied in the literature in the context of bio-cementation, our artificial material nevertheless appears suitable for representing a cemented granular material. Being easy to implement, it could thus enable the calibration of discrete cohesion models for simulation of practical applications.

Appendix A: Theory for the adhesion force of a solid bond

This section proposes a theoretical calculation for the tensile force required to separate a bridge from a sphere by considering an adhesive rupture between the two solid bodies in contact considered both as perfectly rigid. Therefore, this approach consistently differs from the JKR type calculations, adapted to soft bond contacts [31, 34, 41], and is in line with the original analytical elements developed by Ingles [13] for a rigid bond and with more recent modelings [37, 38, 42, 56].

The calculation is conducted with the following assumptions, which are consistent with the materials and experimental conditions studied here.

• First, the mechanical load is assumed to be small enough so that the elastic deformation of the solid bridge can be neglected. This hypothesis will be verified afterwards. Note also that the deformation of the particles is still much lower when they are stiffer than the bridge, as is the case here for glass particles ($E_g\sim 65-70$ GPa) compared to paraffin wax ($E_p\sim 100-200$ MPa).

• Second, considering two perfectly rigid bodies A and B sharing a common contact surface $d\Sigma$, we will assume that the elementary force $\textbf{dF}$ needed to perpendicularly debond the interface simply reads $\textbf{dF} = \sigma _{AB}\mathbf {d\Sigma }$, where $\sigma _{AB}$ is the corresponding adhesive strength between solids A and B, also sometimes called adherence [18].

• Third, the shape of a solid bridge is represented by a simplified geometry made of a cylindrical lateral envelope at the region of contact between the two bonded particles considered as spheres. Moreover, all the bridges are assumed to be strictly identical while the particles they connect are supposed to be exactly in contact. Therefore, the volume of a bridge is that of a cylinder to which it is necessary to subtract 2 spherical caps as shown on Fig. 23. Considering the mode of preparation of our samples, this implies that there is a perfectly uniform distribution of the liquid paraffin within the glass beads. We will further consider that the totality of the introduced paraffin is ultimately located in the bridges, neglecting any coating on the surface out of the contacts. This appears certainly too schematic but nevertheless rather compatible with the observed visualizations (i.e. no paraffin layer is visible on the surface of the beads, even in the micro-tomographic scans).

Fig. 23

Sketch of an idealized paraffin bond

From the theoretical framework of the second hypothesis and introducing the adhesive strength $\sigma _{gp}$ between glass and paraffin, the force required to detach a paraffin bridge (parameterised by the angle $\theta _b$ as shown in Fig. 23) from the surface of a glass sphere of radius R is obtained by integration over the contact surface:

$$\begin{aligned} F_t= & {} \int _{0}^{\theta _b} \int _{0}^{2\pi } \sigma _{gp} R^2 \sin \theta \cos \theta d\theta d\varphi \nonumber \\= & {} 2\pi \sigma _{gp} R^2 \left[ \frac{\sin ^2\theta }{2}\right] =\pi \sigma _{gp} R^2 (\sin \theta _b)^2 = \sigma _{gp} \Sigma _b \end{aligned}$$

(A1)

where $\Sigma _b$ is the cross section of the cylindrical bridge whose radius is $r_b = R\sin \theta _b$. One can alternatively express the $F_t$ in a dimensionless form:

$$\begin{aligned} \overline{F_t}= \frac{F_t}{\sigma _{gp} \pi R^2} \end{aligned}$$

(A2)

Note that a very similar expression than (A1) is expected for a cohesive rupture, i.e. when the rupture occurs within the paraffin bridge itself. One can indeed consider that the failure surface is almost normal to the z-axis, and its area would therefore correspond approximately to $\Sigma _b$. Then $\sigma _{gp}$ needs to be replaced by the intrinsic cohesive strength of paraffin, $\sigma _{pp}$, which is usually refers to as the ultimate tensile strength. For full consistency, the relation $\sigma _{gp} < \sigma _{pp}$ must be verified.

The relation between the paraffin volume content $\xi _p$ and the bond angle $\theta _b$ is obtained relying on the third assumption, considering that liquid paraffin is exclusively located within the bonds, the latter being all identical, and that the glass spheres connected by each bond are in actual contact. The shape of a bond is consequently a cylinder minus two spherical caps and, after minor calculations, the corresponding volume $V_b$ reads:

$$\begin{aligned} V_b = \frac{2\pi }{3} R^3 (1-\cos \theta _b)^2 (1+2\cos \theta _b) \end{aligned}$$

(A3)

This expression can also be written in a dimensionless form as:

$$\begin{aligned} \overline{V_b} = \frac{3V_b}{4\pi R^3} = \frac{1}{2} (1-\cos \theta _b)^2 (1+2\cos \theta _b) \end{aligned}$$

(A4)

In a given sample, denoting by N the number of spherical glass beads (of same radius R) with a mean coordination number Z, the total number of bonds is $\frac{Z}{2} N$. Then, we can determine in the sample the volume of paraffin $V_p$ and glass $V_g$, respectively:

$$\begin{aligned} V_p= & {} \frac{Z}{2} N V_b \end{aligned}$$

(A5)

$$\begin{aligned} V_g= & {} N \frac{4}{3}\pi R^3 \end{aligned}$$

(A6)

This gives the following expression for the paraffin volume content $\xi _p$:

$$\begin{aligned} \xi _p = \frac{V_p}{V_g} = \frac{Z}{2} \overline{V_b} \end{aligned}$$

(A7)

Considering $\overline{F_t}$ and $\overline{V_b}$, there is no obvious one-to-one relationship. However, the following combination can be used:

$$\begin{aligned} \frac{\overline{F_t}^2}{\overline{V_b}} = \frac{2\sin ^4\theta _b}{(1-\cos \theta _b)^2(1+2\cos \theta _b)} = 2 + \frac{2\cos ^2\theta _b}{1+2\cos \theta _b} \end{aligned}$$

(A8)

Then, since $\theta _b$ cannot much exceed $\frac{\pi }{6}$ (i.e. limit case of a pendular bridge in a locally ordered arrangement of spheres), we can use the following approximation for $\theta _b \in \left[ 0,\frac{\pi }{6}\right]$:

$$\begin{aligned} \frac{\overline{F_t}^2}{\overline{V_b}} = 2.61 \pm 0.06 \end{aligned}$$

(A9)

This minor approximation finally allows the following explicit semi-theoretical law to be proposed for the adhesive bond force with an accuracy of $\pm 1.2\%$:

$$\begin{aligned} F_t = \sigma _{gp} 1.62 \pi \sqrt{\frac{2}{Z}} \sqrt{\xi _p} R^2 \end{aligned}$$

(A10)