Particle finite element analysis of large deformation and granular flow problems

Abstract

A version of the Particle Finite Element Method applicable to geomechanics applications is presented. A simple rigid-plastic material model is adopted and the governing equations are cast in terms of a variational principle which facilitates a straightforward solution via mathematical programming techniques. In addition, frictional contact between rigid and deformable solids is accounted for using an approach previously developed for discrete element simulations. The capabilities of the scheme is demonstrated on a range of quasi-static and dynamic problems involving very large deformations.

Introduction

Although most geotechnical structures operate in the small deformation range, there are numerous problems within geotechnical engineering and related fields that call for methods where changes to the problem geometry as a result of deformations are taken into account. These include problems of landslides and debris flow, penetration of various devices such as cones and torpedo anchors into the ground, and the interaction, during installation or under operating conditions, of off-shore oil and gas infrastructure with the seabed.

For some of these problems, the deformation pattern resembles a fluid flow more than a solid undergoing large distortions. In the framework on the standard finite element method, such problems give rise to two fundamental challenges. The first one relates to geometry in the sense that the magnitude of the deformations is bound to lead not only to severe mesh distortion, but also to situations where the boundaries of the problem change from one time step to the next. Of these two separate but related issues, the former has received by far the most attention. Indeed, for many problems the original boundaries are maintained even after relatively large distortions. The perhaps most common approach to avoiding or alleviating mesh distortion is the Arbitrary Eulerian Lagrangian (ALE) method. This method utilizes the respective advantages of pure Eulerian and pure Lagrangian formulations and has been used quite successfully in geotechnical applications [1], [2] and other solid as well as fluid mechanics problems [3], [4]. Another popular method for geotechnical applications is the so-called Remeshing and Interpolation Technique with Small Strain (RITSS) technique proposed by Hu and Randolph [5], [6]. While both the ALE and the RITSS have been used to solve problems involving relatively large deformations, they both have shortcomings in the case where the original boundaries change in the course of the deformation process, for example in the case where an initially contiguous solid separates into two or more parts as a result of external actions.

The second challenge, which in many ways is the more serious one (though it remains much less explored), is that of solving the governing equations – comprising momentum balance, strain–displacement relations and constitutive relations that usually are highly nonlinear and may give rise to ill-posedness, localization of deformations, etc. Indeed, it is well known that even small deformation problems are hard to deal with for constitutive models that involve nonassociated flow rules [7].

In this paper a new scheme that addresses both of the fundamental challenges described above is presented. The scheme is applicable to general large deformation problems with no real limitations on the magnitude of the deformations. In other words, both problems where the deformation patterns resemble fluid flows and those that merely involve the deformation of solids slightly beyond the small deformation limit can be handled. More specifically, issues related to geometry are handled by means of the Particle Finite Element Method (PFEM) [8], [9], [10] while the solution of the governing equations is addressed by means of variational and mathematical programming methods that have their origin in computational limit analysis [11], [12], [13], [14], [15], [16] but since have been applied to a wide range of other problems including elastoplasticity [17], [18], [7] and discrete element type analysis [19], [20], [21]. Other issues dealt with include dynamics and frictional contact.

The paper is organized as follows. In Section 2, the fundamentals of the PFEM are briefly described with emphasis on the alpha-shape method used for identifying solid and void domains on the basis of a cloud of points. Section 3 details the governing equations and their variational formulation. Next, in Section 4, the discretization and solution of the governing equations are described before the treatment of contact is detailed in Section 5. Finally, in Section 6, a number of examples demonstrating the capabilities of the scheme are presented before conclusions are drawn in Section 7. While all aspects of the new scheme are applicable to the general three-dimensional setting, the examples given in this paper are limited to two dimensions assuming plane strain.

Standard matrix notation is used throughout with bold upper and lower case letters denoting matrices and vectors respectively.