Experimental validation of a micromechanically based compaction law for mixtures of soft and hard grains

In this Letter, we report on an experimental study which analyzes the compressive behavior of two-dimensional bidisperse granular assemblies made of soft (hyperelastic) and hard grains in varying proportions ($\kappa$ is the portion of soft grains). By means of a recently developed uniaxial compression setup [Vu and Barés, Phys. Rev. E 100, 042907 (2019)] and using an advanced digital image correlation method, we follow, beyond the jamming point, the evolution of the main mechanical observables, from the global scale down to the strain field inside each deformable grain. First, we validate experimentally and extend to the uniaxial case a recently proposed micromechanical compaction model linking the evolution of the applied pressure $P$ to the packing fraction $\varphi$ [Cantor et al., Phys. Rev. Lett. 124, 208003 (2020)]. Second, we reveal two different linear regimes depending on whether the system is above or below a crossover strain unraveling a transition from a discrete to a continuous-like system. Third, the evolution of these linear laws is found to vary linearly with $\kappa$. These results provide a comprehensive experimental and theoretical framework that can now be extended to a more general class of polydisperse soft granular systems.

Figure 1

(a) Experimental setup. A bidimensional bidisperse granular system, composed of soft and rigid particles, lies on the top glass of a flatbed scanner. A uniaxial compression device stresses it stepwisely while it is imaged from below by the scanner. (b) Composite view of measured fields. Rigid particles are colored in blue. Raw image, in gray, is shown on the left; particle boundaries are in red. Von Mises strain field $\mathcal{C}$ is shown on the right with a color scale going from dark red (low value) to yellow (high value). Contacts are shown in white. (c) Evolution of the measured global force $f$ as a function of the global strain $\varepsilon$ for different softness ratios $\kappa$.

Figure 2

(a) Evolution of the coordination $Z$ relative to the coordination at jamming $Z_J$ as a function of the distance to the jamming transition for the packing fraction $\varphi -{\varphi }_J$. The dashed line corresponds to a power law with an exponent 0.5. (b) Evolution of the global stress applied to the system $P$ as a function of $\varphi -{\varphi }_J$. Plane line curves correspond to Eq. (1). In both panels, the different curves correspond to different softness ratios $\kappa$ given in (a).

Figure 3

(a) Linear evolution of the soft particle average asphericity $\langle a\rangle$ as a function of the global stress applied to the system $P$. Their slope is given in the inset as a function of the softness ratio $\kappa$. (b) Linear evolution of the average value of the relative contact length $\langle l\rangle$ as a function of the distance to the jamming transition for the packing fraction $\varphi -{\varphi }_J$. In both panels, the different curves correspond to different $\kappa$, given in the legend in (b).

Figure 4

(a) and (b) Evolution of the global stress $P$ and of the relative packing fraction $\varphi -{\varphi }_J$ as a function of the average value of the von Mises strain $\langle \mathcal{C}\rangle$ in soft particles for different $\kappa$ values, given in the legend in (b) (for the three panels). In (a) and (b), straight black dashed lines show slopes of 2.2 kPa and 5.4, respectively, as a guide for the eyes. The vertical dashed line and the shaded area show the crossover strain ${\mathcal{C}}_c=0.991\pm 1.2\times {10}^{-3}$ and its error bar, respectively. It splits the space horizontally between small and large deformation levels. (c) Evolution of the standard deviation of the von Mises strain in soft particles as a function of the relative packing fraction $\varphi -{\varphi }_J$. The straight dashed line with a slope of $4.0\times {10}^{-4}$ is a fit of the curves collapsed in their linear regime. These curves leave their linear regime at certain cutoff whose approximated values are given in the inset as a function of the softness ratio $\kappa$.