Applications in Geomechanics

The application of the Boundary Element Method to the linear three-dimensional soil-structure interaction problem is discussed. Detailed formulations for rigid, surface, massless foundations of arbitrary shape are given in both frequency and time domains. In both cases the foundations are assumed to rest on a linear elastic, homogeneous, and isotropic half-space and are subjected to either externally applied loads or obliquely incident body or surface waves. Results obtained by the above approaches as well as by other well established techniques are given in a comparison study. More general problems involving massive foundations and superstructures are also presented in the general framework of a substructure formulation.

Dynamics of foundations is part of the more general field dynamic soil-structure interaction, which is concerned with the study of structures based on flexible soils and subjected to dynamic actions that may be directly applied to the structure or transmited through the soil.

The mechanical behavior of the soil is characterized by the presence of the fluid phase. Indeed, the interaction between the soil skeleton and fluid phase affects the behavior of soil in various situations. This interaction is especially important when the applied compression squeezes the fluid out of the soil. This phenomenon, or what was called ‘consolidation’ originally, was so typical as a subject in soil mechanics that the name ‘consolidation analysis’ has almost become a generic term for any soil deformation computations which take the effect of fluid into account. It is in this sense that we use the word ‘consolidation’ in this chapter.

The Boundary Element Method (BEM) has been used for solving saltwater-intrusion problems for about six years. This paper attempts to review the use of the BEM for various hydrologic situations, which are described by different mathematical models: 1) steady-state cross sectional flow of freshwater above a stagnant saltwater zone; 2) transient cross sectional flow with the horizontally moving interface; 3) transient cross sectional flow with the vertically moving interface; and 4) transient horizontal flow, simplified by the use of the Dupuit-Forchheimer and Ghyben-Herzberg approximations. A brief discussion of each model’s assumptions is also included. BEM models are concluded to cover a broad spectrum of saltwater-intrusion problems and to be superior to the finite-difference method (FDM) and the finite-element method (FEM) models for the moving boundary (interface) problems. The BEM seems also to be more efficient for analyzing horizontal-flow models, as long as the Ghyben-Herzberg approximation can be used. However, the FDM and the FEM are still better tools for analyzing the regional, transient saltwater-intrusion problems, for which this approximation cannot be used.

In this article we examine the application of the boundary element method to the study of the non-linear interface behaviour between two material regions. The non-linear interface response is modelled either by Coulomb frictional behaviour or by interface plasticity. An incremental formulation is adopted for the analysis of the non-linear pheonomena. The incremental non-linear analysis is used to examine the two-dimensional problem of a finite elastic region which contains a circular rigid inclusion. The numerical results presented in the paper illustrates the manner in which the non-linear phenomena at the inclusion-elastic medium interface contributes to the global non-linear responses in the composite.

Of late, the Boundary Element Method (BEM) has emerged as a serious contender in the computational mechanics arena. It has been successfully applied to numerous engineering problems. The general perception of the BEM is that it is inherently more efficient because it is a boundary, instead of a domain, scheme. It is also probably more accurate because Green’s function, which is admissible to the governing equation, is used as weighing functions. The major shortcoming of the BEM, however, is its relative inability in dealing with nonlinearities and heterogeneities, which are frequently encountered in engineering. To overcome the difficulties, some authors introduced variations, in which domain discretization and/or integration are usually needed, of the genuine boundary method. Although ways have been devised to optimize the domain treatment, the inherent efficiency of the boundary method can be seriously hurt.

Boundary element method is employed to analyze the effects of spacial distribution of soil infiltration properties and rainfall rate on the hydrologic performance of catchment area. For simplicity these are modeled by recharge problem to the groundwater flow. The simulation model also can determine hydrograph bias due to variability of soil infiltration properties and rainfall rate in space. Further, the hydrograph bias due to movement and time variation of recharge source is also studied.

The general equation governing unconfined groundwater flow is a transient, three-dimensional partial differential equation with variable coefficients. As summarized in this chapter, for many problems the equation can be simplified such that the associated free space Green’s function is known. Then the Boundary Element Method (BEM) becomes a very efficient technique for solving for the location of the transient free surface.