## All Theses

8-2010

Thesis

#### Degree Name

Master of Science (MS)

#### Legacy Department

Mechanical Engineering

Joseph, Paul F

#### Committee Member

Thompson , Lonny L

Li , Gang

#### Abstract

A general method based on the singular integral equations is developed to
computationally determine the higher order coefficients in mixed mode fracture
mechanics. These 'k' and 'T' coefficients are defined with respect to a polar coordinate
system centered at a crack tip, and give asymptotic expressions for stresses and
displacements according to the William's eigenfunction expansions,
(r, ) (2r) k f (n, ) k f (n, ) IIk
ij
II
n
Ik
ij
I
n
n 0
2
1
n
ij      

1
(0, ) (2 ) ( , ) ( , ) , , ; , , I IT n I IT II IIT
ij n ij n ij
n
T f  r T f n  T f n  i r  j r 

      
In the above expression the n = 0 terms correspond to the modes I and II stress
intensity factors and the so called, T-stress. From a method point of view, the higher
order k-coefficients are easily obtained, while the T-coefficients require significant postprocessing
of the singular integral equation solution. A planar crack parallel to an
interface between two elastic materials and subjected to far-field tension is considered as
an example and extensive results are presented. This example is chosen due to the
anomalous behavior of a closing crack tip as the crack approaches the interface for
certain material combinations. Such 'Comninou contact zones' occur even in a tensile
field when the crack is within a critical distance from the interface. Numerous results are
provided that compare the asymptotic solutions with that of the full-field. It is shown that
up to four k-coefficients and many T-coefficients can be determined for h/a = 0.001,
where h is the distance of the crack from the interface and a is the half-crack length.
While the application of the method to the case of a crack parallel and very close to an
interface focuses on the anomaly of a closing crack tip, in general the ability to determine
iii
higher order coefficients can be used to quantify the size of the zone in which linear fracture mechanics is valid.

COinS