X-ray 3D imaging-based microunderstanding of granular mixtures: Stiffness enhancement by adding small fractions of soft particles

Abstract

This research focuses on performing ultrasound propagation measurements and micro-X-ray computed tomography (µXRCT) imaging on prestressed granular packings prepared with biphasic mixtures of monodisperse glass and rubber particles at different compositions/fractions. Ultrasound experiments employing piezoelectric transducers, mounted in an oedometric cell (complementing earlier triaxial cell experiments), are used to excite and detect longitudinal ultrasound waves through randomly prepared mixtures of monodisperse stiff/soft particles. While the fraction of the soft particles is increasing linearly from zero, the effective macroscopic stiffness of the granular packings transits nonlinearly and nonmonotonically toward the soft limit, remarkably via an interesting stiffer regime for small rubber fractions between 0.1 ≲ ν ≲ 0.2. The contact network of dense packings, as accessed from µXRCT, plays a key role in understanding this phenomenon, considering the structure of the network, the chain length, the grain contacts, and the particle coordination. While the maximum stiffness is due to surprisingly shortened chains, the sudden drop in elastic stiffness of the mixture packings, at ν ≈ 0.4, is associated with chains of particles that include both glass and rubber particles (soft chains); for ν ≲ 0.3, the dominant chains include only glass particles (hard chains). At the drop, ν ≈ 0.4, the coordination number of glass and rubber networks is approximately four and three, respectively, i.e., neither of the networks are jammed, and the chains need to include particles from another species to propagate information.

Keywords: X-ray computed tomography; granular materials; particles’ contact network; small-strain stiffness; ultrasound waves.

Fig. 1.

P-wave modulus plotted against the fraction of rubber particles for samples compressed uniaxially from p= 40 kPa to 200 kPa. Each rubber fraction was tested five times with the SDs shown as error bars for p = 80 kPa and 160 kPa (others are not shown for the sake of visibility).

Fig. 2.

Number of particles N_p carrying a particular number of contacts c for (A) the entire sample, (B) the glass subnetwork, and (C) the rubber subnetwork, extracted from µXRCT scans for samples with different rubber fractions under uniaxial compression at p= 160 kPa.

Fig. 3.

Glass and rubber networks of samples prepared with ν= 0.1, (A) glass network, (B) rubber network, and ν= 0.5, (C) glass network, (D) rubber network, both under p= 160 kPa compression. Different colors represent the numbers of contact each particle carries. Particles carrying zero, one, two, three, four, five, or above are colored black, magenta, blue, red, green, yellow, and orange, respectively.

Fig. 4.

The coordination number of different networks shown in Fig. 2. The dashed line is ${\overline{C}}_J=$ 5.5 represents a jamming regime of a packing with friction equal to one.

Fig. 5.

One of the identified shortest percolating paths of samples with different rubber fractions, ν= (A) 0.1, (B) 0.2, (C) 0.3, (D) 0.4, (E) 0.5, and (F) 0.6, under p = 160 kPa compression. To indicate the shortest identified percolating path, green and yellow are used to distinguish further between glass and rubber particles. Red and blue indicate glass and rubber particles in the remaining sample area apart from the percolating path.

Fig. 6.

(A) Schematic view of the in situ µXRCT setup with a three-quarter section view of the oedometer cell. (B) Segmented 3D reconstructed sample consisting of 50% glass and rubber particles with identified particles centroids and branch vectors, i.e., a vector connects centers of two neighboring particles.