## University of Wollongong Thesis Collection 1954-2016

2004

#### Degree Name

Doctor of Philosophy

#### Department

School of Mathematics and Applied Statistics - Faculty of Informatics

#### Abstract

In this thesis, we study the advanced boundary element method for fracture mechanics, including static and dynamic problems. Static problems are solved by using the dual boundary element method, that the dual equations are the displacement and the traction boundary integral equations. An efficient integral equation formulation is proposed with the displacement equation being used on the outer boundary and the traction equation being used on one of the crack faces, which bases on the works in the paper Dual boundary integral formulation for 2-d linear elastic crack problems (Submitted to Journal of Computational Mathematics). Discontinuous quarter point elements are used to correctly model the displacement in the vicinity of crack tips. Using this formulation a general crack problem can be solved in a single-region formulation, and only one of the crack faces needs to be discretised. Once the relative displacements of the cracks are solved numerically, physical quantities of interest, such as crack tip stress intensity factors can be easily obtained. Normally, the stress intensity factor is obtained by using discontinuous quarter point element method. Because quadratic boundary elements do not correctly describe the behaviour of displacement near the crack tips, special crack tip elements are needed to model the displacement in the vicinity of crack tips. In this work, we present a special crack tip element method, which provides similar accuracy as that of quarter point element method, but a much easier discretisation of the crack face for evaluating stress intensity factors. Further, a new subregion boundary element technique is presented to solve composite material problems, which bases on the work in the paper A new subregion boundary element method (Published by 15th International Conference on Boundary Element Technology 2003). Similar composite problems are also solved by using domain decomposition method, which based on the work in the paper A new subregion boundary element technique based on the domain decomposition method (Submitted to Engineering Analysis with Boundary Elements). The technique is more efficient than traditional methods because it significantly reduces the size of the final matrix. This is advantageous when a large number of elements need to be used, such as in crack analysis. Also, as the system of equations for each subregion is solved independently, parallel computing can be utilized. Further, if the boundary conditions are changed the only equations required to be recalculated are the ones related to the regions where the changes occur. This is very useful for cases where crack extension is modelled with new boundary elements or where crack faces come to contact. Dynamic fracture mechanics problems are solved by using the dual reciprocity boundary element method, which based on the work in the paper A subregion DBEM formulation for dynamic analysis of two dimensional cracks (Accepted by Mathematical and Computer Modelling). The dual reciprocity boundary element method employing the step by step time integration technique is developed to analyse two-dimensional dynamic crack problems. In this method the equation of motion is expressed in boundary integral form using elastostatic fundamental solutions. In order to transform the domain integral into an equivalent boundary integral, a general radial basis function is used for the derivation of the particular solutions. The dual reciprocity boundary element method is combined with an efficient subregion boundary element method to overcome the difficulty of a singular system of algebraic equations in crack problems. Dynamic stress intensity factors are calculated using the discontinuous quarter point elements.

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