Applying the Drucker-Prager Failure Criterion for Representing Soil Behavior using Smoothed Particle Hydrodynamics

Smoothed Particle Hydrodynamics (SPH) has previously used in hydrodynamics as a Lagrangian numerical method to simulate fluid behavior. Recently the SPH is also being used to simulate soil deformation, since the largely known Finite Element Method (FEM) cannot represent the soil deformation after failure. In SPH, soil materials are modeled as a set of particles, and the behavior of those particles can be simplified using the equivalent viscosity or using the soil failure criterion. This research tries to implement the Drucker-Prager model using Fortran platform to simulate a simple soil model deformation behavior, by modifying the previous model that used the equivalent viscosity. A simple model of small slope is built by a stack of SPH particles, which are expected to behave as the collapsing soil mass. The movement of the particles and the change of the model’s geometry are observed visually after the SPH simulation. The preliminary results show that the particles can already behave like a failure in granular soil, yet it still needs to be improved due to the unwanted particle movements.

Fortran; Numerical method; Smoothed particle hydrodynamics; Soil constitutive model

The most popular numerical method used in geotechnical analyses nowadays is the Finite Element and Finite Difference. Those methods have the ability of simulating the stress-strain behaviour of soil, hence they can predict the deformations. However, both numerical methods have drawback regarding the large deformation or the deformation after failure due to the mesh and grid system. To overcome this problem, the Smoothed Particle Hydrodynamics had been introduced.

 Smoothed Particle Hydrodynamics (SPH) is a mesh-free or gridless particle method based on Lagrangian formulation, that was first applied to solve the astrophysical problems in open space (e.g., Gingold and Monaghan, 1977; Lucy, 1977). Recently, this method is also applied for geomaterials and geodisaster models as mentioned by Huang et al. (2014) and Bui and Nguyen (2021). Examples of applications are for dam-break analysis (Wang and Shen, 1999), large deformation and slope failure (Bui et al., 2011), seepage (Maeda et al., 2004), piping erosion (Sjah and Vincent, 2017), liquefaction and flow failure (Naili et al., 2005; El Shamy and Sizkow, 2021; Sizkow and El Shamy, 2021). This article presents the progress of the on-going research on the use of SPH model to simulate the soil deformation after failure. The large lateral deformation phenomenon after the big earthquake in Central Sulawesi, Indonesia, in 2018 highlights the importance of the prediction of post failure deformation. The SPH research group in the Department of Civil and Environmental engineering of University of Indonesia tried to develop their own platform to model the post-liquefaction deformation. The first attempt has been conducted using the equivalent viscosity as in (Mahardima et al., 2021). The liquefied soil behaviour was modelled as Bingham fluid, which is one of the viscoplastic models (e.g., Uzuoka et al., (1998).

In this article, the Elasto-plastic SPH Procedure is used in its programming algorithm with the Drucker-Prager model to describe the behaviour of soil particles which refers to research that has been developed by Bui et al. (2008). The Drucker-Prager failure criterion is one of the soil constitutive law to represent the elasto-plastic soil behaviour, and is considered as the most stable compared to the other criterion (e.g., Potts and Zdravkovi?, 2001). The recent model is built base on the previous codes by Mahardima et al. (2021) using Fortran platform. This study is a small part of the bigger research scheme with the main objective is to build a SPH model to predict the large deformation after liquefaction.

      This article shows the progress of an on-going research of application of SPH method to the simulation of granular materials using the Fortran platform. The main objective is to model the behavior of granular materials (i.e., sands) while interacting with water during the liquefaction phenomena. At the recent study, the results have shown that the proposed algorithm can simulate the behavior of granular material as particles after the failure. Yet the interaction between particles still not completely correct, possibly due to the later part of the algorithm, which is the collision handling. This part will be the next subject for the next study.

        The authors of this article want to acknowledge the SPH Research Group at the Department of Civil Engineering University of Indonesia for the helps and discussions during this study. This research was also supported partially by The Ministry of Education, Culture, Research, and Technology through the WCR Research Grant NKB-389/UN2.RST/HKP.05.00/2021.

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