ABSTRACT

In this research, we investigated the deformation field near a crack tip due to
thermal stress. Crack tip singularities are examined using line integral that exhibits
path independence of all contours near the tip of the crack in a two dimensional
deformation field of an elastic material. The approach of using the J- Integral is
used to extract the magnitude of crack tip stress intensity factor for thermal stress
crack problem. The characteristics of the J-integral in the presence of thermal
stress are determined and a process of accessing the crack tip stress intensity
factors is developed.

TABLE OF CONTENTS

Cover page …………………………………………………………(i)
Title page ………………………………………………………….( ii)
Approval Page ………………………………………………….(iii)
Dedication …………………………………………………………(iv)
Acknowledgement ………………………………………………….(v)
Abstract …………………………………………………………(vi)
Table of Content ………………………………………………….(vii)
CHAPTER ONE
INTRODUCTION…………………………………………………………….(1)
CHAPTER TWO
REVIEW OF RELATED LITERATURE ………………………(4)
CHAPTER THREE
J – INTEGRAL …………………………………………………………..(19)
CHAPTER FOUR
ANALYSIS …………………………………………………(28)
CHAPTER FIVE
DISCUSSION, SUMMARY AND CONCLUSION………………..(45)
REFERENCES ………………………………………………… (46)

CHAPTER ONE

INTRODUCTION
In contemporary times, the society is challenged with the problem of fracture
as long as there are structures made by man, and this limitation could be as a result
of anomalies created in the course of structural design which culminate into failure
in systems. This failure has been traced to high degree of stress concentration in a
giving region of a body particularly solids thereby leading to deformation and crack.
Collapse in solids could be fatal and have greatly affected lives, properties and
finances negatively in nations of the world. Cracks lower the structural integrity of
structures. Beyond that, due to load bearing material property above and beyond
conventional capacity is required to examine fracture resistance of a solid. When
stress applied to solids is large enough, distortion, reformation of atomic bonds
occur and move to a new position and the original material deform. Permanent
deformation may take place under certain external load which is mainly due to
activation of slip of system such as dislocation. However, when plastic deformation
is very large, the contribution of elastic strain becomes negligible. Many engineering
structures have cracks and we often see that some of them have failed around a small
crack. When a material is under loading beyond its capacity crack is bound to occur
and this failure is related to stress field in the neighborhood of crack tip as result of
deformation. So, there are stress and strain that take place inside real bodies.
2
In this research, we investigate the deformation field near a crack tip due to
thermal stress. This analysis is of great significance in field of Fracture Mechanics in
which crack tip singularities are examined using line a integral that exhibits path that
is independent of all contours near the tip of a crack in a two dimensional
deformation field of an elastic material. The principle of continuous deformation
path is related to path independence in which deformation the path consist of only
points at which a function is analytic where the integral retain the same value.
According to Rice [1] and Amazigo [2], this integral provides not only an accurate
characterization of the crack tip elastic-plastic field but also a good elastic fracture
criterion. Therefore, there is an arbitrary contour that will yield the same value for
different paths. Elastic deformation and irreversible dislocation flaw begin with the
very first small increment of load and the dislocation flaw accelerates thereafter with
increasing load up until the ultimate rupture.
Failure in materials caused by engineering structures that contain cracks is a
menace to mankind. Normally, any structure that is containing crack should fail no
matter how small the crack or how light the load. The act of predicting failure of
these structures becomes paramount. In this work, we concentrate on the elastic
analysis of deformation field in the neighborhood of a crack tip under thermal stress.
Advances in Fracture Mechanics have helped in understanding the mechanism of
failure thus, helping to prevent structural failure and their frequencies.

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