## Theses and Dissertations

2015

Dissertation

#### Degree

Doctor of Philosophy

#### Department

Mechanical Engineering

Nied, Herman F.

Delph, Terry J.; Webb, Edmund; Pearson, Raymond A.

#### Abstract

The bi-material crack problem is an interesting and important topic in the field of fracture mechanics. The existing mainstream solutions, either analytical or computational, are commonly focused on some specific cases, e.g., a crack lying on exactly the bonded border of dissimilar materials, or a crack impinging upon a bi-material interface at a right angle. However, little attention is paid to the general cases, i.e., cracks approaching or attacking the material divided border arbitrarily, which is more likely to happen in the engineering products. With any possibility of the crack's incidence angle, the asymmetric nature of the geometry and the materials property induces more difficulties in the mathematical formulation of the crack-tip stress field. The conventional analytical methods may not be a convenient way for the derivation, especially of the fracture parameters. For this end, in this study, the Williams' expansion method is exploited to investigate the two-dimensional/three-dimensional fracture problem in which the crack terminates at a biomaterial interface with an arbitrary angle of incidence. The characteristic equation is obtained and solved to investigate the distribution of dominant roots. Mathematically, a matrix-based system is developed, which can be easily used to formulate the general asymptotic solution of the singular stress and displacement fields surrounding the crack-tip. The theory of singularities is introduced to represent the mixed-mode nature of the solution for the arbitrarily-oriented crack. This concept is further employed for the cases with complex singularities. After that, the relationship of the asymptotic field and the linear elastic fracture parameters is established directly through a linear system. In addition, taking advantage of the enriched element approach, the derived formulation in this study is programmed and implemented in a finite element analysis. This provides an efficient and effective method for simulating and solving different types of crack problems, especially with complicated geometries, loading patterns and material combinations. Then different mixed-mode fracture criteria for predicting the direction of crack growth are introduced. With the method discussed in this study, the maximum circumferential stress criterion is considered to be the most appropriate one, but needs to be slightly modified for multiple material problems. Finally, some examples of numerical solution of the asymptotic fields are demonstrated using the computed stress intensity factors and the developed matrix system for the general crack cases with an arbitrary impinging angle with respect to an interface. The numerical results for specific cases are compared with the existing references.

COinS