Crack initiation equation

However in the past days, the creations of the new machines were not considering one of the material weaknesses in their application which bring them into failure. The failure of the materials did a lot of damage and lost while some of the failures were also bring fatality to humans. Basically, fatigue of materials occurs as they were used repeatedly subjected to reverse stresses over the time. When the materials reach its maximum cycles, it no longer can hold the load as it did before.

Due to fatigue, the material fails and bring catastrophe to the mankind and also to the surroundings. Some of the catastrophes that involved fatality due to fatigue were two Comet jet planes were crashed due to metal fatigue. The first crash killed 29 passengers and a crew lost their lives.

Three months later, the second crash killed fourteen passengers and seven crews British Broadcasting Corporation. However, there was 34 years later, another airline catastrophe really did change the airline industry related to the material control.

Since then a lot of actions were taken to prevent any other catastrophe that happens due to fatigue. Huge number of engineers made the research on metal fatigue due to dynamic loading. Some of the tests were also involved the study of crack initiation which is then lead the further stage of fatigue failure.

This focus area is done based on the following aspects: i AISI steel is used as the material that to be studied. In this chapter, the readers will get chances to understand the idea on the importance of failure analysis in the engineering field. Starting from section 2. In the same section, the readers also will be introduced with an overview about finite element analysis FEA including the steps of FEA. Section 2. This section will bring the readers to the relations of fatigue damage value and crack initiation life cycles in order to precede this project into the analysis.

The following sections will show the brief of the specimen geometry 2. The last section will give the summary of this chapter in proceeding to the next chapter of this project. Some of the most used metal type in the engineering design is iron and its alloys which are also known as steel.

It has accounted for huge production of metals and becoming the most favourite metal chosen by the engineers mainly because of the combination of good strength, toughness, and ductility at low cost. In smaller portions of steel types, there are variety of steel family such as plain-carbon steels, alloy steels, stainless steels, cast iron and copper alloys. They are widely used in manufacturing of various types of parts and tools, which are having the needs of high rotation and force acted on them.

Carbon steel can be classified according to various deoxidation practices, which have effects on the steel properties. Variations of the carbon content in the steels also give a huge effect on the mechanical properties; with increasing the carbon content will lead to the increase the strength and hardness of the steel. A , — Hernandez-Gomez, I.

Sauceda-Meza, G. Hou, X. Jin, X. Fan, et al. Mousavi, M. Aliha, and D. Seitl, P. Miarka, and V. Jenq and S. Saghafi, M. Ayatollahi, and M. A 21—22 , — Ayatollahi, and J. Rock Eng. Aliha, G. Hosseinpour, and M. Roy, R. Narasimhan, and P. Qian and W. Kim and K. Kotowski, G. Lesiuk, J. Correia, and A. Fatigue 23 , — Ren, M. Ulbin, and J. Mirsayar, A. Razmi, M. Aliha, and F. Qian and A. Ren, Z. Zhu, Q. Check out our fatigue crack growth calculator based on the methodology described here.

Cracks commonly occur in engineered parts and can significantly reduce their ability to withstand load. Cracks typically form around pre-existing flaws in a part. They usually start off small and then grow during operational use. A crack in a part will grow under conditions of cyclic applied loading, or under a steady load in a hostile chemical environment. Crack growth due to cyclic loading is called fatigue crack growth and is the focus of this page. Crack growth in a hostile environment is called environmental crack growth and is not discussed here.

The analysis of fatigue crack growth relies on the concepts of fracture mechanics which are discussed on this page. If you are not familiar with fracture mechanics, it is recommended that you read that page before proceeding. A typical plot showing the growth of a crack is provided below. The crack size, a , is shown as a function of cycles, N , of applied load. Notice that the crack initially grows very slowly, but the growth accelerates i. The reason for this acceleration in growth is that the growth rate is dependent on the stress intensity factor at the crack tip , and the stress intensity factor is dependent on the crack size, a.

As the crack grows the stress intensity factor increases, leading to faster growth. The crack grows until it reaches a critical size and failure occurs.

The figure below shows stresses applied in a cyclic manner. For simplicity, the figure shows constant values of maximum and minimum stresses; however, the loading in a realistic scenario may be much less uniform and may consist of multiple sets of stress ranges. The stress ratio is an important quantity, and is the ratio of the minimum stress to the maximum stress:.

Recall that the stress intensity factor is a function of geometry and applied stress:. The values Y and a are dependent on geometry, so for a specified crack and part geometry, the stress intensity factor is proportional to applied stress. Therefore, we can define the following:.

Note that if the definitions for maximum and minimum stress intensity are substituted into the definition for the stress intensity range, a new, useful definition for stress intensity range can be obtained:. Another useful relationship can be derived by combining the equations for stress intensity range and R-ratio:.

When a cyclic load is applied to a material, the stress intensity range is calculated as discussed in the previous section :. A fluctuating stress intensity drives the crack to grow at some rate. The crack growth rate in a material takes the form shown in the figure below.

It increases the work that is necessary to propagate a crack as the work now includes not only the surface energy of the crack-faces that are created but also the energy consumed in plastically deforming the matrix in the vicinity of the crack-tip. The second effect is to relax the stress concentration at the tip to blunt the crack. The plastic deformation at the root of the crack would also blunt the tip of the crack and increase r, the radius of the crack, which thus increases the fracture stress.

The x-ray studies have confirmed the presence of heavily distorted layer immediately beneath the fracture surface. The modified form of equation is-. More and more strain energy is released as the crack length 2c increases, though a part of this energy is used in forming the surfaces of the cracks. The remaining energy is changed into the kinetic energy. As a crack propagates, the material at the sides of the crack moves apart with a finite velocity. Kinetic energy is associated with this movement of the material near the end of the crack.

As the size of the crack becomes more than half critical crack length, the crack gains speed, and the velocity approaches a maximum value 0. A crack can move in a brittle material at the velocity of sound in that material. Inorganic glasses contain inherently some bulk flaws, but the cracks which lower the strengths of the glasses probably are on the surface such as fine scratches, which may be introduced in the manufacturing process.

Moreover, a surface crack of depth c is equivalent to an interior crack of length 2c. The initiation of fracture in polymers too is associated with prior presence of the flaws. Griffith theory has been verified by experiments on glasses and polymers at low temperatures, where a simple process of fracture by the propagation of elastic cracks occurs i.

Most industrial ceramics contain porosity and flaws of varying sizes, and thus the strength varies from specimen to specimen. There is no reliable evidence to prove that Griffith cracks exist in metals in the unstressed condition. However, even when a metal fails by brittle cleavage, a certain amount of plastic deformation almost always occurs prior to fracture. Enough experimental evidences are available to prove that micro-cracks can be produced by plastic deformation. Micro-cracks can form at non-metallic inclusions or brittle second phase in steels by decohesion from matrix, or by cracking of the particle as a result of plastic deformation as illustrated in Fig.

The experiments have shown that cracks responsible for brittle-cleavage type fracture are produced by plastic deformation.

Plastic deformation to produce pile-up of dislocations at an obstacle. Such an obstacle could be an inclusion, or second phase particle or even grain boundary or twin interface or at the junction of two intersecting slip planes. Because slip is often observed even when the specimen fractures in a brittle manner, it is clear that slip dislocation must play some part in the fracture process. Micro-crack-initiation occurs due to buildup of shear stress at the head of the dislocation pile-up.

The formation of a crack in a carbide plate can initiate cleavage fracture in adjacent ferrite if stress concentration is relatively high. It has been seen if the dispersed second-phase particle is readily cut by the dislocations, then at some obstacle large pile-up of dislocations occurs. It leads to the development of high stresses, easy initiation of micro-cracks, and the brittle fracture.

In crystalline materials, as some plastic deformation always occurs, then it is the slip dislocations generated immediately prior to fracture which give rise in certain regions of the crystal to micro-cracks, even without the presence of brittle particles.

Several models have been put formed for the process whereby slip dislocations are converted into micro-cracks. Zener was the first who advanced the idea that the high stresses produced at the head of a dislocation pile-up could produce the fracture as illustrated in Fig.

A second mechanism of crack formation, suggested by Cottrell is the one arising at the junction of two intersecting slip planes as illustrated in Fig. Here two slip dislocations combine to form an edge dislocation. The new dislocation having a Burgers vector of a [], is a pure edge dislocation. This dislocation may be considered as a wedge. Even intersecting deformation twin bands have been seen to create cracks. All BCC-metals show increase of yield stress with the fall of temperature, that is, the stress to move the dislocation, the Peierls-Nabarro stress increases with the fall of temperature.

But as the velocity of dislocation-motion is proportional to the stress, the first dislocations move very rapidly with the fall of temperature. Thus, the dislocation squeezing as illustrated in Fig.