Experiments

Chapter III: Interparticle forces and effective stress in unsaturated granular

3.3 Experiments

𝑀_𝑎= 𝜌𝑉_𝑎 = (1−𝑆_𝑟)𝜙 𝜌𝑉 . (3.34) With Eqs. (3.31), (3.33), and (3.34), Eq(3.32) can be rewritten as:

𝜕

𝜕 𝑡 (𝑉_𝑎

𝑉 )= 𝜕

𝜕 𝑡 {𝛽^𝑎

1(𝜎₀−𝑃_𝑎) +𝛽^𝑎

2(𝑃_𝑎−𝑃_𝑤)}

=−

𝐷 𝑅_𝑔𝑇 𝑚_𝑎(𝑃_𝑎+𝑃)

𝜕²𝑃_𝑎

𝜕 𝑦²

+ (1−𝑆_𝑟)𝜙 𝑃_𝑎+𝑃

𝜕 𝑃_𝑎

𝜕 𝑡

. (3.35)

The partial differential form of Eq. (3.35) can then be written as follow:

𝜕 𝑃_𝑎

𝜕 𝑡

=−𝐶^𝑎

𝜕 𝑃_𝑤

𝜕 𝑡 +𝑐^𝑎

𝑣

𝜕²𝑃_𝑎

𝜕 𝑦²

(3.36) 𝐶^𝑎 = (𝑃_𝑎+𝑃)𝛽^𝑎

2

(𝑃_𝑎+𝑃) (𝛽^𝑎

1− 𝛽^𝑎

2) + (1−𝑆_𝑟)𝜙 𝑐^𝑎

𝑣 =

𝐷 𝑅_𝑔𝑇 𝑚_𝑎{(𝑃_𝑎+𝑃) (𝛽^𝑎

1− 𝛽^𝑎

2) + (1−𝑆_𝑟)𝜙} where 𝐶^𝑎 is the interactive constant of the air phase, and 𝑐^𝑎

𝑣 is the coefficient of consolidation of the air phase. The excess pore water pressure and excess pore air pressure can be calculated by numerically solving the two PDEs Eqs. (3.30) and (3.36) simultaneously.

packing.

Figure 3.5: The experimental setup to validate the stress partition equation Eq.

(3.10). (a) The container for partially saturated granular packing: 1. piston cap, 2.

piston, 3. transparent window, 4. U-shaped sidewall, 5. support, 6. nylon screws, washers, and hex nuts. (b) EVA plastic tubing fixed in the gap on the side wall for pressurization and depressurization. (c) Pressure sensor to measure the pore air pressure. (d) CCD cameras to track the deformation of the grain skeleton and the geometry of pore water clusters. (e) Custom made valve for the depressurization process. (f) A simple illustration of the 1D consolidation experiment during which an external load𝜎₀is applied to the granular system through the piston. The pore air depressurization is controlled with a custom made valve, while a pressure sensor is used to measure pore air pressure in real-time.

The container, as shown in Fig. 3.5a, includes two transparent windows, a U-shaped sidewall, a piston, a piston cap, and two supports. Polycarbonate plates (100mm x 100mm x 6.35mm) were used as transparent windows to facilitate optical imaging.

The inner surface of the two windows was coated with hydrophobic ceramic spray to prevent pore water from infiltrating into the gap between the particles and the window. The side wall was machined out of white UHMW polyethylene plates.

The piston, the piston cap, and the supports were made of Delrin. Additionally, nylon socket head screws were used to hold all the parts together. In order to seal the experiment setup during pressurization, grease and sealing tapes were applied to the piston and side walls.

The schematic of the one-dimensional consolidation experiment is depicted in Fig.

3.5f. While particles are confined in the container, a dead load𝜎₀is applied through the piston to the granular system, resulting in an excess pore air pressure. During the consolidation process, the start of the depressurization of the pore air phase is

controlled by the custom made valve (Fig. 3.5e). The pore air pressure is measured by the pressure sensor (Fig. 3.5c) that connects to the container using an EVA plastic tubing (Fig. 3.5b). In addition, the deformation of the particles and geometry of the fluid clusters are tracked using CCD cameras (Fig. 3.5d). The particle strain field is therefore computed using digital image correlation analysis (DIC), and the pore fluid pressure is calculated using the Young-Laplace equation (Eq. 3.16), which will be discussed in the following section.

The particles of the granular packings studied in the experiments consist of soft neoprene rubber disks with two different sizes (Fig. 3.5a). These disk-shaped particles are assumed to be linear elastic and incompressible (Poisson’s ratio≈0.5) with Young’s modulus 𝐸 measured to be 0.8 MPa. Additionally, the Coulomb friction coefficient was determined to be 𝜇 ≈ 0.4 after wetting. The water contact angle𝛼of both large and small particles is measured to be around 70°. The granular packings used in the experiments contain 12-13 large particles (diameter = 15 mm, thickness = 1.5 mm) and 10-12 small particles (diameter = 9.5 mm, thickness = 1.5 mm). To create an appropriate grayscale pattern for DIC analysis [35], white multi-surface paint was sprayed on all the rubber disks to generate speckle patterns.

DIC analysis was performed using the xyz code [36].

Tracking and analysis of fluid clusters

The fluid used in the experiment is water stained with carmine red dye. After the particles are placed in the container, the dyed water is slowly injected into the granular packing using a syringe and a flexible thin needle. The position and geometry of the water clusters at each time point are recorded with a color CCD camera (Nikon AF Nikkor 50mm). The precise geometry and contour of the water clusters are further extracted using a Matlab color threshold program. The image data from the two cameras are combined by pairing the four corners of the imaging area. While the position and geometry of the solid disks are determined by DIC, the solid-water contact surfaces on a particle can be found by choosing a proper distance threshold 𝑅^∗ (𝑅₀ < 𝑅^∗ < 1.05𝑅₀, 𝑅₀is the radius of the solid disk). As shown in Fig. 3.6, for a solid particle𝑖, if the distance between the centroid of𝑖and any point on the contour of water cluster 𝑗 is smaller than the threshold𝑅^∗, then the particle 𝑖 is considered to be in contact with water cluster 𝑗. All the points on the contour of water cluster 𝑗 that fall in this threshold form the solid-water contact surface𝑆

𝛽 𝑖. Therefore, the corresponding filling angle 𝜃^𝛽

𝑖 as well as the average normal vector

¯

𝒏 can be easily computed using Matlab. Similarly, the boundary-water contact

surfaces can be determined by choosing another boundary distance threshold 𝑅^∗

𝑏

(0< 𝑅^∗

𝑏 < 0.08𝑅₀).

Figure 3.6: Schematic of finding the average curvature𝐻_𝑓, average normal vector ¯𝒏, and fluid-gas pressure differenceΔ𝑃^𝛽 at a fluid-gas contact surface of fluid cluster 𝜔_𝑗.

After finding the solid-water contact surfaces, the water-air interfaces of the water cluster 𝑗 are segments on its contour that do not belong to a solid-water contact surface or a boundary-water contact surface. While there may be multiple water-air interfaces on water cluster 𝑗 that could be used to calculate the pressure difference Δ𝑃^𝛽, for the precision of computation, only the longest contour segment (largest distance between the two endpoints) is utilized here. The two endpoints of the segment as well as the point on the contour segment that is closest to the midpoint of the two endpoints are chosen. The radius of curvature𝑅is then calculated by fitting a circle using these three points. Finally, the water surface tension𝛾is assumed to be constant throughout the experiments, and the pressure differenceΔ𝑃^𝛽 is calculated using the Young-Laplace equation (Eq. (3.16)).