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# A Quasi-Static Method for Large Deformation Problems in Geomechanics

P. A. Vermeer, L. Beuth, T. Benz
Institute of Geotechnical Engineering, University of Stuttgart, Germany

Keywords: large deformations, meshfree methods, Material Point Method, slope failure
ABSTRACT: The Finite Element Method (FEM) has become the standard tool for the analysis of a wide range of mechanical problems. However, the classical FEM is not well suited for the treatment of large deformation problems since excessive mesh distortions require remeshing. The Material Point Method (MPM) represents an approach in which material points moving through a fixed finite element grid are used to simulate large deformations. As the method makes use of moving material points, it can be classified as a meshfree method. With no mesh distortions, it is an ideal tool for the analysis of large deformation problems. All existing MPM codes found in literature are dynamic codes with explicit time integration and only recently implicit time integration. In this study, a quasi-static MPM is presented. The paper starts with the description of the quasi-static governing equations, the numerical discretisation and an explanation of calculation procedures. Afterwards, geotechnical boundary-value problems are considered.

• Introduction
When problems involving large deformations are modelled with an Updated Lagrangian FEM, considerable mesh distortions occur, which require computationally time-consuming remeshing. To overcome the difficulties of the FEM, meshfree methods have been developed, e.g. the Element-Free Galerkin Method and Smoothed Particle Hydrodynamics (Li et al., 2002). The Material Point Method might be classified as a meshfree method or an Arbitrary Lagrangian-Eulerian method (Więckowski, 1999). The early beginnings of the MPM can be traced back to the work of Harlow (Harlow, 1964), who studied  fluid flow by material points moving through a fixed grid. Sulsky et al. (1996) later extended the approach for the modelling of solid mechanic problems and called it the Material Point Method. Bardenhagen et al. (2000) extended the method further to include frictional contact between deformable solid bodies. The potential of the MPM for simulating granular flow was first recognised by Więckowski (1998). Several papers on the MPM modelling of granular flow were published (Więckowski et al., 1999; Więckowski, 1998 and 2003). Coetzee (2004) and Coetzee et al. (2005) extended the method to include a micro-polar Cosserat continuum and described applications to anchor pull-out and excavator bucket filling. MPM uses two discretisations of the continuum, one based on a computational mesh and the other based on a collection of material points or “particles”. All the properties and state parameters of the continuum as well as external loads are carried by the material points, while the grid carries no permanent information. The computational grid is used to determine incremental displacements of material points by solving the governing equations as with the standard FEM. Large deformations are modelled by memorizing incremental deformations through material points that move through the mesh. By this approach, MPM combines the advantages of both Eulerian and Lagrangian formulations.

Most MPM implementations developed so far are dynamic codes, which employ an explicit time integration scheme. Although it is possible to use these programs also for the analysis of quasi-static problems, this is computationally inefficient, as explicit integration requires very small time steps. For these reasons, it was decided to develop a quasi-static MPM implementation, which uses an implicit integration scheme, thereby broadening the possibilities of large deformation analyses for complex, large-scale geotechnical problems. Meanwhile, the new quasi-static code has been validated (Beuth et al., 2007) by considering some  boundaryvalue problems with known analytical solutions. In fact, excellent agreement was found between results obtained with the MPM and reference solutions. The method is now being adapted and applied to solving geotechnical problems like slope failures or, at a later stage, pile penetrations. For the calculation of the presented problems, high-order elements with quadratic interpolation of displacements have been employed as they reproduce stress

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