| In the first part, a new analytical approach, within the extended finite element (XFEM) framework, is proposed to compute Strain Energy Release Rates (SERRs) directly from Irwin's integral. Crack tip enrichment functions in XFEM allow for evaluation of integral quantities in closed form (for some crack configurations studied) and therefore results in an accurate and efficient method. The effects of high order enrichments, mesh refinement and the integration limits of Irwin's integral are examined in benchmark numerical examples. The results indicate that high order enrichment functions have significant effect on the convergence, in particular when the integral limits are finite. When the integral limits tend to zero, simpler SERR expressions are obtained and high order terms vanish. Nonetheless, these terms contribute indirectly via coefficients of first order terms.;The analytical formulation is then extended to cracks in arbitrary orientations. Several benchmark examples are investigated including off-center cracks, inclined cracks and crack growth problems. On all these problems, the method is shown to work well, giving accurate results. Moreover, due to its analytical nature, no special postprocessing is required which leads to a fast approach to obtain Strain Energy Release Rates. Thus it is concluded that this method may provide a good alternative to the popular J-integral method.;In the second part of the thesis, the stress-strain behavior of short single walled carbon nanotube (SWCNT) aggregates is investigated by a novel incremental constrained minimization approach. An AIREBO potential is used to model the interactions within and between CNTs. The idea is to homogenously disperse SWCNTs in the computational cell at random positions and orientations following spherical uniform distributions, and incrementally deform the cell while restraining the movement of atoms at the ends of nanotubes. The stress-strain response of the system is obtained in each loading direction and it is shown to converge to an isotropic behavior (a similar response in all directions) as the number of CNTs in the system increases. In addition, it is shown that the Young's modulus of the system increases linearly with the CNT aggregates density and the method agrees well with results obtained from molecular dynamics simulations running at near zero degrees kelvin, which are obtained at only a fraction of the CPU time required for MD methods. |
