Elements with penalized equilibrium and rotational degrees of freedom in fracture mechanics problems

Abstract

This thesis deals with the solution of linear elastic fracture mechanics problems. To solve the linear elastic fracture mechanics problems, the finite element method and path independent integrals are employed, namely Rice's J integral and an alternative path independent integral I *, which is the energy complement to J. Stress intensity factors for typical mode I and mode II fracture mechanics problems in isotropic and orthotropic elastic plates are calculated. The problems considered are a center cracked panel subjected to uniform tension, a single edge cracked panel subjected to uniform tension, a double edge cracked panel subjected to uniform tension, and a center cracked panel subjected to uniform shear. Firstly, classical displacement based finite elements, elements with penalized equilibrium and elements with drilling degrees of freedom are presented and implemented in a MATLAB environment. Secondly, two different ways to evaluate the stress intensity factor are considered, namely the displacement extrapolation approach, and the path independent integrals J and I *. The numerical implementation and path independence of the J and I * integral is demonstrated. It is shown that the J integral can estimate the lower bound of the stress intensity factor when used with displacement based finite elements, while the I * integral can estimate the upper bound of the stress intensity factor, when used with stress equilibrium elements. Thirdly, the path independent integrals J and I * are applied to isotropic fracture mechanics problems to determine the stress intensity factor at the tip of a crack. Convergence studies are presented to investigate the influence of mesh refinement on the stress intensity factor predicted using the J and I * integral. The path independence of J and I * are investigated. Numerical results for typical fracture specimens are presented and discussed. Finally, the path independent integrals J and I * are applied to orthotropic fracture mechanics problems to determine the stress intensity factor at the crack tip. Again, convergence studies are done, and the path independence of J and I * are investigated for orthotropic problems. Numerical results for typical fracture specimens are presented and discussed. The effect of the degree of anisotropy and fiber orientation on the stress intensity factor is also demonstrated. A novel contribution in this thesis are the results for elements with drilling degrees of freedom in fracture mechanics problems. In addition, the results presented here may serve to clarify published stress intensity factor results for orthotropic materials presented in the literature, since many of the results previously presented are contradictory.