Multiscale models are a powerful alternative to continuous and discrete models in the numerical study of granular materials. In the 3D H model [6], the mesoscale is composed of a ten-grain hexagonal cell whose symmetries enable to have an analytical incremental constitutive relationship, and to characterize the geometry of the cell with only one angle. The Representative Elementary Volume (REV) at the macroscale is a spatial distribution of the cells. The 3D H model capacities were demonstrated on drained triaxial tests performed on sand [7].

As the model has been developed and validated for purely frictional contact, it can only be used for the modelling of dry or saturated soil. The model has to be extended in order to be used in unsaturated conditions. The purpose of this work is to establish a relationship between the volume of water, the geometry of the cell and the capillary forces in the cell.

The capillary forces in a bridge between two spherical grains has been characterized, using Laplace-Young equation [1,3] or empirical relationship [4]. However, the study of more complex regime of water between several grains resulting of bridges coalescence is still an open subject [2,5], in particular for systems with more than three grains. Our work consisted in performing calculations with the energy minimizer software Surface Evolver on various hexagonal cell configurations. The software enables to define a simple initial geometry with boundary conditions at the liquid/grains interfaces and to determine the geometry with the lowest energy by a gradient descent method. The capillary forces can be deduced from the energies of close configurations by the principle of virtual work.

This approach is validated on a pendular bridge between two grains and compared to results obtains in the literature [1]. Then, the coalescence of two bridges in a three grains configuration is studied and compared to experimental data [2]. Finally, the capillary forces are calculated in the H model cell, for different capillary regimes.

This study aims to validate the use of energy minimizing method with two and three grains in order to develop an empirical relationship between the volume of water, the geometry of the cell and the capillary forces. The perspectives of this work is to implement these results in the 3D H model for unsaturated soil modelling.

References

[1] Duriez, J. & Wan, R. Contact angle mechanical influence in wet granular soils. Acta Geotechnica, Springer Nature, 2017, 12, 67-83

[2] El Korchi, F. Z. Approche expérimentale multi-échelle de l'effondrement capillaire de sols granulaires. PhD thesis, Université de Montpellier, 2017.

[3] Lian, G.; Thornton, C. & Adams, M. A theoretical study of the liquid bridge forces between two rigid spherical bodies. Journal of Colloid and Interface Science, 1993, 161, 138-147

[4] Richefeu, V.; Radjai, F. & El Youssoufi, M. S. Stress transmission in wet granular materials. The European Physical Journal E, 2006, 21, 359-369

[5] Urso, M.; Lawrence, C. C. J. & Adams, M. A two-dimensional study of the rupture of funicular liquid bridges. Chemical Engineering Science, 2002, 57, 677-692

[6] Xiong, H. Multiscale modelling of granular materials in application to geotechnical engineering problems. PhD thesis, Université Grenoble Alpes, 2017.

[7] Xiong, H.; Nicot, F. & Yin, Z. Y. A three-dimensional micromechanically based model. International Journal for Numerical and Analytical Methods in Geomechanics, Wiley-Blackwell, 2017, 41, 1669-1686