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Material Point Method for Large Deformation Modeling in Geomechanics Using Isoparametric Elements


The geotechnical engineering field is full of large deformation problems, ranging from settlements caused by human constructions to naturally occurring landslides. Modeling these problems is therefore of paramount importance for risk management and improved understanding of soil/rock behavior. Many particle-based or mesh-free methods have been developed to reliably and efficiently analyze large deformation problems. One such method is the Material Point Method (MPM) (Sulsky et al., 1994; 1995), which is a hybrid Eulerian-Lagrangian approach capable of simulating large deformation problems of history-dependent materials. While the MPM can represent complex and evolving material domains by using the Lagrangian material points, boundary conditions are often applied to the Eulerian nodes of the background mesh. Consequently, the use of a structured mesh may become prohibitively restrictive for modeling complex boundaries such as a landslide topography.

This research explores the suitability of unstructured background mesh with isoparametric elements to model irregular boundaries in the MPM. An inverse mapping algorithm is used to transform the material points from the global coordinates to the local natural coordinates. Dirichlet velocity and frictional boundary conditions are applied in the local coordinate system at each boundary node. This approach of modeling complex boundary conditions is validated by a patch test, modeling of a gravity-driven rigid block sliding on an inclined plane, and performing mesh convergence study on the stress obtained in a hollow cylinder problem with pressure applied on the outer boundary.

The proposed approach is then applied to a sand production problem. 2D MPM with isoparametric elements is used to simulate two different sets of laboratory experiments labeled the PEA-135 and PTC which investigate sand production phenomenon on Castelgate Sandstone. The material points representing the material volume (or mass) are allowed to be removed, representing the disintegration of sandstone rock to sand. This removal is dictated by a criterion developed based on the equivalent plastic deviatoric strain accumulated. To the author's best knowledge, this is the first sand production study using the MPM. A new constitutive model, the Bonded Norsand, is developed to accurately describe the behavior of the sandstone material. Calibration of the parameters are done through available laboratory triaxial drained compression tests on the material.

This method is later applied to a flume test of controlled dry granular flow on an inclined plane conducted by the United States Geological Survey (USGS). This flow problem provides an opportunity to demonstrate the ability of the proposed approach to model large deformation problems with high velocity or kinetic energy in contrast with the sand production problem. Mesh sensitivity and time step effects are investigated. The approach is able to capture most of the mechanisms of dry granular flow. Comparisons are also made to other approaches within the MPM, as well as other continuum-based methods such as the Finite Element Method (FEM) and Smooth Particle Hydrodynaics (SPH).

Lastly, an expansion of this approach to 3D is presented where the inverse-mapping is solved iteratively and demonstrated with three examples. The first is the 3D version of the previously discussed flume test of controlled dry granular flow on an inclined plane. The second is an experiment performed on the same flume with the same dry sand but with a slit gate to demonstrate the ability to model variations in 3D. Lastly, the approach is used to model a historic debris flow, the Yu Tong Road landslide in Hong Kong. The approach is found to be able to simulate the mechanisms of these different cases. Comparisons with the results from other continuum-based methods such as the Arbitrary Lagrangian-Eulerian approach and the SPH are also made in this section.

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