Investigation of the effect of specific interfacial area on strength of unsaturated granular materials by X-ray tomography

Abstract

This paper studies the effect of interfacial areas (air–water interfaces and solid–water interfaces) on material strength of unsaturated granular materials. High-resolution X-ray computed tomography technique is employed to measure the interfacial areas in wet glass bead samples. The scanned 3D images are trinarized into three phases and meshed into representative volume elements (RVEs). An appropriate RVE size is selected to represent adequate local information. Due to the local heterogeneity of the material, the discretized RVEs of the scanned samples actually cover a very large range of degree of saturation and porosity. The data of RVEs present the relationship between the specific interfacial areas and degree of saturation and gives boundaries where the interfacial area of a whole sample should fall in. In parallel, suction-controlled direct shear tests have been carried out on glass beads and the material strength has been corroborated with two effective stress definitions related to the specific air–water interfacial areas and fraction of wetted solid surface, respectively. The comparisons show that the specific air–water interfacial area reaches the peak at about 25% of saturation and contributes significantly to the material strength (up to 60% of the total capillary strength). The wetted solid surface obtained from X-ray CT is also used to estimate Bishop’s coefficient χ based on the second type of effective stress definition, which shows a good agreement with the measured value. This work emphasizes the importance to include interface terms in effective stress formulations of unsaturated soils. It also suggests that the X-ray CT technique and RVE-based multiscale analysis are very valuable in the studies of multiphase geomaterials.

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Acknowledgements

This work was funded by F.R.S-FNRS of Belgium with Project No. PDR.T.1002.14. The first author also appreciates the support of Taishan Scholar Project of Shandong Province of China during paper writing and revision.

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Appendices

Appendix 1: 3D image segmentation

The region growing method has four steps. Firstly, as seen in Fig. 12, a partial thresholding is applied to the gray values. Half of the air phase and half of the solid phase are selected based on the first and third peak values in the histogram (voxels with a gray level smaller than the peak value of 20, noted as \(P_{\text{a}}\), are air and voxels with gray level larger than the peak value of 140, noted as \(P_{\text{g}}\), are solid). For the middle phase, which is water, the thresholding is taken around the middle peak to select part of the water phase. As the amount of voxels of water is varying with water content (the peak is not obvious when the degree of saturation is low), the selection is based on the sample with 46.7% degree of saturation. We take voxels with gray level larger than 49 and less than 58 (noted as \(P_{\text{w1}}\) and \(P_{\text{w2}}\), respectively) as the water phase, of which the probability density of the two gray level values are 80% of the middle peak at 46.7% of degree of saturation. Secondly, a procedure is taken to consider the Partial Volume Effect (PVE). Voxels on the boundary of the solid phase may have both air and water phases, but the gray level may fall in the range of the water phase thresholding. To filter the PVE, a sphere with 2 voxels radius around each voxel is selected. The voxels in the spheres on the solid–air interface have higher variance in gray-level distribution and the spheres inside the water phase have lower variance. A filtering procedure is applied to reset the voxels on the solid–air interface to undefined voxels by thresholding an appropriate variance. Thirdly, a simultaneous region growing process is applied to assign the neighboring undefined voxels into a phase. The voxels which are neighbors of only one phase with the gray-level values falling in a tolerance threshold of a phase are attributed to that phase. The tolerance range is set as: air phase < \(P_{\text{w1}}\), \(P_{\text{a}}\) < water phase < \(P_{\text{g}}\), solid phase > \(P_{\text{w2}}\). Step by step, each phase will grow up until it meets with another phase. At the interface, when the undefined voxels are neighbors of more than one phase, the voxels are set to be undefined until the final step. At the final step, the undefined voxels on the interfaces are assigned to a phase based on its most present neighboring phase. The final procedure is repeated until all voxels are attributed to the three phases. After the four steps, the 3D reconstructed images are segmented.

Fig. 12
figure12

Thresholds of the region growing method by Hashemi et al. [20]

Appendix 2: Suction-controlled direct shear test

The suction-controlled direct shear test device is depicted in Fig. 13a. The tested glass beads are placed in two superposed plexiglass cylinders. The bottom one is fixed, while the top one can be moved horizontally, driven by a weight (a cup of sand in our case) suspended to a nylon rope through a pulley. The weights of the added sand and the cup, therefore, provide the horizontal shear force. The normal stress is controlled by a top lid in which a certain number of counterweights can be added. The normal force acting on the shear plane is the total weight of the counterweights, the lid, the top cylinder and the part of the sample above the shear plane. For dry or wet samples, the normal stress on the shear plane can be calculated as:

$$\sigma_{\rm n} = \frac{N}{A} + \rho_{\text{s}} gh\left( {1 - n} \right) + S_{\text{r}} n\rho_{\text{w}} gh$$
(17)

where \(N\) is the normal force on the shear plane except the sample itself, \(A\) is the cross-sectional area, \(\rho_{\text{s}}\) is the density of the solid grains, \(g\) is gravity, \(h\) is sample depth above the shear plane, \(n\) is sample porosity, \(S_{\text{r}}\) is the degree of saturation, and \(\rho_{\text{w}}\) is the density of water.

Fig. 13
figure13

The suction-controlled direct shear test setup. a Layout of the device; b control of suction and measurement of degree of saturation

There is a small circuit at the bottom of the lower plexiglass cylinder, allowing water imbibition and drainage. A porous stone is placed at the bottom of the sample to distribute water more homogeneously over the cross section of the sample. The small circuit is connected with a burette through a plastic tube, and there is a valve at the end of the burette. Dry glass beads are poured into the cylinder to prepare a dry sample. The weight of the two cylinders is measured before and after glass beads filling, which gives the mass of the dry sample. After the two cylinders are filled with glass beads, the lid is put on the top of the sample and a certain load is applied. The procedure to apply suction is presented in Fig. 13b. Firstly, the valve is opened to allow water flow into the dry sample until some water comes out from the top. Then, the water level in the burette is adjusted to be the same as the top of the upper cylinder. This process saturates the sample. Then, the burette is lowered to a certain level. In this procedure, some water may flow out of the sample into the burette. After the stabilization of the water level, the water volume change in the burette, \(z_{2}\), is the water volume drained out from the sample. Combining with the sample size and the solid phase mass, the current degree of saturation in the sample can then be calculated. The level difference between the water table in the burette and the shear plane, \(z_{1}\), gives a negative pressure in the shear plane by multiplying with water density and gravity (\(\rho gz_{1}\)). Therefore, suction in the sample is \(\rho gz_{1}\) and the pore air pressure is 0 since the top of the sample is connected with atmosphere from the edge of the lid. After the application of normal load and suction, sand is poured into the cup on the other side of the pulley gradually until failure occurred. Then, by dividing the total weight of the cup and sand with the area of the horizontal cross section, the failure shear strength can be obtained.

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Wang, JP., Lambert, P., De Kock, T. et al. Investigation of the effect of specific interfacial area on strength of unsaturated granular materials by X-ray tomography. Acta Geotech. 14, 1545–1559 (2019). https://doi.org/10.1007/s11440-019-00765-2

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Keywords

  • Effective stress
  • Interface
  • Strength
  • Unsaturated granular material
  • X-ray tomography