Abstract:
An efficient numerical procedure for modeling planar cracks in three dimensional, linear elastic, infinite medium which accounts for the influence of surface stresses is presented in this dissertation. The concept of surface stresses, which has been widely employed in the investigation of nano-scale problems, is adopted to derive a suitable mathematical model capable of simulating nano-sized cracks. An infinitesimally thin layer of material on the crack surface is modeled by a zero-thickness surface perfectly bonded to the bulk material, with its behavior governed by the Gurtin-Murdoch constitutive relation. In the formulation, the classical theory of isotropic linear elasticity is utilized to establish the governing equation of the bulk material in terms of singularity-reduced boundary integral equations for the displacement and traction on the crack surface. For the zero-thickness surface, the final governing equation incorporating the surface stress effect is obtained in a weak form following the standard weighted residual technique. The fully coupled system of equations is then solved by the FEM-SGBEM coupling numerical procedure. Due to the weakly singular nature of all involved boundary integral equations, standard continuous interpolation functions can be employed everywhere in the approximation of crack-face data and only special quadrature for evaluating nearly and weakly singular integrals is required. Once the implemented numerical scheme is validated with available benchmark solutions, it is applied to investigate the nano-scale influence on nano-sized cracks. Several examples are presented to demonstrate the capability and robustness of the method. Results from an extensive parametric study reveal that, the presence of surface stresses not only increases the near-surface material stiffness but also introduces size dependent behavior of solutions and the reduction of stresses in the region ahead of the crack front.
