| Since composite and laminated materials have been prevalently applied in many engineering areas, both the robustness and reliability of multi-material solids are paid more and more attention in the aspect of mechanics. The fracture problem with cracks located near the bonded interface of a bi-material composite has been considered an important issue since the 1960's. A lot of previous research work focused mainly on particular cases, such as a crack along or perpendicular to the bonded interface. However, cracks in the composite material may not always have a specific orientation relative to the interface. An investigation of cracks with any possible incidence angle would be more meaningful in practical engineering applications.;In this study, the general characteristic equation of the bi-material fracture problem, containing a 2-D semi-infinite isotropic crack meeting the bonded interface with an arbitrary angle, is derived using the Williams' series approach for both in-plane and anti-plane cases respectively. A new type of geometric configuration is utilized due to the asymmetry of material properties, stress and displacement fields. The general characteristic equations have been reduced to several specific cases to show its validity. A system of boundary conditions given in matrix form is used to solve for the eigenvalues, resulting in either twelve unknown coefficients for the in-plane case or six coefficients for the anti-plane case.;As another contribution of this study, a revised self-adaptive Newton-Raphson iteration method is formulated to numerically compute dominant roots of the characteristic equation. It has been found that for the in-plane case, a second eigenvalue exists, rather than the conventional single root. The asymptotic contribution from double roots should improve the accuracy and convergence of numerical solutions for asymptotic analyses. The eigenvalues switch between two complex and two real numbers at certain transition points depending on the material combination. The distributions of roots, imaginary parts and real parts, for varying crack orientations are displayed graphically and discussed. Some features of the numerical roots for anti-plane problems are also shown. In this case, only a single real eigenvalue occurs for any material or crack angle cases.;Ultimately, finite element method is combined with asymptotic theory. The implementation of finite element analysis is performed using the academic software FRAC2D for fracture analysis, is introduced in this study. Taking advantage of enriched finite elements, the important fracture parameters, i.e., stress intensity factors, are calculated directly as DOFs during the numerical analysis. A new formulation, based on linear algebraic transformations, is described to derive the asymptotic terms for the displacement and singular strain fields in the form of Williams' series expansion.;Finally, the derived solutions in this study were embedded into the FRAC2D program and tested on some simple 2-D examples involving an isotropic finite crack perpendicular to the bi-material interface. The new version of software works well and the results show a perfect agreement with other available solutions. |
