Modeling of secondary cracks using branching theory by boundary element method

Dissertation to receive a master's degree

Stone Mechanical Engineering

Abstract

The unexpected and stress intensity factor of some engineering structures has caused a lot of financial and collateral damage. In investigating the causes of failure, researchers gradually found that the design of many of these structures was correct based on the theory of elasticity and resistance of materials, and the cause of failure was cracks that existed in the structure or were created during work. Therefore, it was concluded that the design and analysis of these structures was not successful based only on the two lessons mentioned. Accordingly, in the second decade of the 20th century, a new science called fracture mechanics [1] was founded, which examines the analysis of structures based on the existence or creation of cracks. Considering that the theory of fracture mechanics examines the stability of structures based on the criteria of crack propagation, including the energy principle and the resistance of objects against cracks, it is possible to obtain a more complete judgment regarding the stability or instability of structures by examining the conditions for crack propagation, how the crack branches or stops. Whenever the energy released due to crack expansion is two or three times the resistance of the crack, the cracks branch into two or three branches. This research has been conducted with the aim of investigating the analysis of secondary cracks using branching theory. For this purpose, by using the obtained equation and the physical conditions of a crack, the first and second branching points of cracks have been obtained for four types of rocks and compared with the results obtained from the theory of fracture mechanics, which has been seen to be in good agreement with the results obtained from the theory of fracture mechanics.

Introduction

Despite the prosperity and comfort that technical knowledge has brought to mankind, unfortunately, the sudden and unexpected failure of some engineering structures has caused a lot of financial and human losses. For example, the report of NASA [1] in 1976, the damages caused by the failure of structures and the efforts to prevent them cost the United States about 119 billion dollars annually. Considering the human lives that were lost in these accidents adds to the importance of this issue

The wide use (but not completely correct) of metals in the 19th century caused many accidents and many victims. For example: In the 1860s-1870s, accidents on the railway lines caused the death of 200 people in England every year. Most of these accidents are due to failure of train wheels, rails or axles. has been 

Incidents have happened during the last 200 years, written by Anderson[2]. Some of these incidents include:

In March 1935, about 700 people gathered to watch a boat race on the Montrose Suspension Bridge[3]. During the race, one of the chains broke and many people died.

On January 22, 1866, a part of the roof of the Manchester railway station fell, causing the death of two people. The cause of the accident was the failure of a cast iron member.

In investigating the causes of failure, the researchers found that the design of many of these structures was correct based on the theory of elasticity and resistance of materials, and the reason for the failure was cracks that existed in the structure or were created during work. Therefore, it was concluded that the design and analysis of these structures was not successful based only on the two lessons mentioned. Accordingly, in the second decade of the 20th century, a new science called fracture mechanics [4] was founded, which examines the analysis of structures based on the existence or creation of cracks. Discovering the flaws and defects in the materials and fixing them will prevent some unfortunate incidents. The emergence of material production methods along with the expansion of material science has brought the number of accidents to a lower and more acceptable level. Most of the failures occurred under low stresses. Research shows that the cause of these failures are defects such as small cracks [5]. [1]

1 – 2 cracks in a structure

Consider a crack in a structure. As a result of repeating intermittent loads or due to a combination of load and environmental effects, this crack grows with time. As the length of the crack increases, more stress concentration is created in it.As the stress concentration increases, the rate of crack propagation increases with time. Crack propagation as a function of time is plotted in Figure (1-1).  Due to the presence of cracks, the resistance of the structure decreases, this resistance may be lower than the resistance for which the structure is designed. As it is evident in figure (1-2), the strength of the structure decreases drastically with the increase of cracks. After a certain period of time, the strength of the structure decreases to such an extent that the structure can no longer withstand high sudden loads and the structure may break due to them. (Images and diagrams are available in the main file) found so that the system breaks under the service load [1].

Some structures are designed for high loads. These loads are large enough to cause cracks in the structure. Especially when there are primary defects or stress concentration in the structure. The designer should take into account the possibility of cracking and as a result the collapse of the structure due to this factor. This point conveys the fact that every structure has a specific life. Of course, this probability of failure during the whole lifetime of the part must be lower than an acceptable limit. In order to achieve this goal, it is necessary to be able to predict the speed of crack development and also the speed of reduction of the residual strength of the structure. Making these predictions (predicting the rate of increase in crack size and predicting the decrease in structural strength) and also developing prediction methods are among the goals of fracture mechanics. Considering Figure 1-1, the fracture mechanics should be able to answer the following questions [1] :

What is the residual strength of a structure as a function of the crack size?

What is the critical crack size in the structure?

How long does it take for a crack of a certain length to reach its critical size? reach?

How long can the primary defects (cracks) be at the beginning of a part's life?

At what time intervals should the structure be examined in terms of crack evaluation?

The science of fracture mechanics has complete answers for some and useful answers for the rest of the above questions.        

The field of fracture mechanics is divided into two general sections [1]:

Materials science examines the fracture process at the scale of atoms and dislocations.

Applied mechanics examines the stress field at the crack tip, elastic and plastic deformation in the vicinity of the crack tip.

For the correct application of failure mechanics in engineering problems, it is necessary to have a general knowledge of each of these parts.

In the resistance of materials, the stress parameter (resistance) represents the stability and tolerance of a member against incoming loads. But since in fracture mechanics it is always assumed that every part has a series of defects and small cracks, the stress parameter alone is not enough to express the tolerance of a member. Because if we consider two members of the same gender and with completely equal dimensions, so that one has a smaller crack and the other contains a larger crack, it is obvious that the latter will break much sooner than the former. Therefore, a parameter should be considered, which represents the length of the crack. This parameter, which is called stress intensity factor, will be examined in detail below.