Boundary element formulations for elastoplastic stress analysis problems
This thesis presents an advanced quadratic formulation of the boundary element (BE) method for two-dimensional elasto-plastic analysis in which 3-node isoparametric quadratic elements are used to model the boundary and 8-node isoparametric quadrilateral quadratic elements are used to model the interior domain. The main objectives of the research are to present a comprehensive review of the many different BE approaches in elasto-plasticity to investigate the potential accuracy, robustness and reliability of each approach, and to implement the favoured approach in a comprehensive computer program for use by engineers. Full details of the elasto-plastic analytical formulations and numerical implementations are presented without ambiguity or omission of details. A brief review of the basic principles of plasticity is presented followed by the expressions for elasto-plastic flow rules and the numerical implementations which treat mixed hardening material behaviour. The analytical BE formulation in linear elastic applications is presented. Full details of its expansion to elasto-plastic problems are shown. Two main BE approaches in elasto-plasticity are presented in detail in this work; the initial strain displacement gradient approach with its compulsory modelling of the partial or full interior domain, and the particular integral approach which can be applied exclusively to the surface avoiding any modelling of the interior. It was decided that the initial strain displacement gradient approach is more robust than the particular integral approach and is more likely to be favoured by an inexperienced user of a BE program, despite its main disadvantage of interior modelling. The initial strain displacement gradient formulation as well as other alternative formulations are presented. The values of stress and strain rates at interior points are calculated via the numerical differentiation of the displacement rates in an element-wise manner; an approach similar to that used in Finite Element (FE) formulations. Full details of the numerical implementation algorithm which uses incremental and/or iterative procedures are presented. The details of the particular integral approach which circumvents the strongly singular integrals arising in domain integrals are also discussed in detail. A computer program for the particular integral approach was written, but, due to the constraints of time and the added complexity of this approach, it was not possible to fully test the program on practical elasto-plastic cases within this project. A full computer program, in Fortran, based on the initial strain displacement gradient elasto-plastic BE formulation is written and applied to several practical test problems. The program is written with emphasis on clarity at the expense of efficiency in order to provide a foundation for extension to three-dimensional applications and more complex plastic behaviour. The BE solutions are compared with the corresponding FE solutions provided by the commercially available FE package, ABAQUS, and, where appropriate, exact analytical solutions. The BE solutions are shown to be in very good agreement with other analytical and numerical solutions. It is concluded that the numerical differentiation of displacement rates in an element-wise manner is an accurate and numerically efficient technique which enables the strongly singular integrals to be performed.