Half-Plane Boundary Element Fundamental Solutions and Body Force

Document Type : Research Article


University of Zanjan


Two-dimensional half-plane fundamental solutions have been developed by different researchers in the fields of electronics, mechanics and geotechnics. However, for geotechnical purposes, their solutions are not complete. This paper discusses those previous solutions and details the mathematical procedures for obtaining a new and complete set of half-plane boundary element fundamental solutions. Initially, static equilibrium equations were written using Papkovitch functions and a proper Green’s function was presented for a two-dimensional half-plane space. Having applied the second Green’s identity, the stress-free condition for the ground surface has been satisfied in the displacement and traction fundamental solutions. These solutions can be applied in a meaningful way to problems with semi-infinite work spaces like those much seen in the geophysics, geotechnical and mining engineering because they do not need to discretize the distal boundaries of the model. After extracting half-plane fundamental solutions, the effects of the gravity force as body force and required functions for a half-plane boundary element analysis were extracted. The effectiveness and accuracy of the new solutions have been evaluated by implementing them in a boundary element computer code and solving several classic semi-infinite examples. Results showed that the new solutions are capable of accurately and economically modeling semi-infinite problems.


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