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Contents
Introduction............................................................3
Chapter One.............................................................6
1.1. From the history of fracture.................................study............................6
1.2. Macroscopic and microscopic view of fracture....................8
1.3. Fracture modes................................................10
1 .4. Review of experimental fracture studies.......................12
1.5. Finite element method.........................................14
1.6. Atomistic computer simulation.................................16
1.7. Review of computer simulation of fracture studies.............17
1.8. Mixed fracture modes..........................................26
Chapter Two.............................................................29
2. The computer simulation model.................................29
Chapter Three...........................................................35
3.1. Computer simulation fracture experiments under tension
deformation (mode I)...........................................35
3.1.1. Deformation interval from 0 to 15%..........................35
3.1.2. Deformation equal to the 15%...........................36
3.1.3. Deformation equal to the 20%...........................38
3.1.4. Deformation equal to the 28%...........................40
3.1.5. Deformation equal to the 34%...........................42
3.2. Observations of the full fracture mode I process stages.........66
3.3. Computer simulation under mixed mode loading (I+II).............89
3.3.1 Applied deformation ratio (si/su) is < 1 ...............90
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3.3.1.1. The simulation result at applied deformation ratio equal to
0.329 90
3.3.1.2. The simulation result at applied deformation ratio equal to
0.353 92
3.3.1.3. The simulation result at applied deformation ratio equal to
0.411 94
3.3.2. Applied deformation ratio (Sj/eu) is >1.....................104
3.3.2.1. The simulation result at applied deformation ratio equal to 17................................................................104
3.3.2.2. The simulation result at applied deformation ratio equal to 28 106
3.3.2.3. The simulation result at applied deformation ratio equal to 30................................................................108
3.4. Fracture simulations of crystal with different crack distance... 122
3.4.1. Crack length equal to 10 atomic spacings.................123
3.4.2. Crack length equal to 30 atomic spacings.................125
3.4.3. Crack length equal to 10 atomic spacings.................127
3.4.4. Crack length equal to 10 atomic spacings.................129
Conclusion...............................................................144
Reference................................................................147
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Fracture studies in solid Ar using computer simulation
One of the most important and key concepts in the entire field of materials science and engineering is fracture. Research efforts in the field of fracture mechanics since early 1950’s have resulted in many applications and use of many parameters to predict the instability condition in a wide spectrum of materials under the influence of load.
Fracture is a very complex process which involves the nucleation and growth of dislocations, micro and macro voids and cracks. From the point of view of mechanics, fracture in its simplest definition; a single body begins to separate into pieces by an imposed stress. From the point of view of microscopic focus, fracture is simply the breaking of atomic bonds between atoms.
For analytical purpose a crack line has been idealized as one-dimensional defect on a flat cleavage plane. That is why there are three modes of cracks corresponding to the different orientations of external stress with respect to the fracture plane and they are distinguished by indices I, II and III.
Many research efforts have been confined to investigation and understanding of both materials and structure response under a mode I loading condition. However, in practice, materials are often subjected to either a mode I/II or mode I/III loading.
As the fracture in materials is very complex process and diverse, it is
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therefore obvious that a detailed understanding of fracture will ultimately require an understanding of processes, taking place at the atomic scale.
Since experimental information can hardly ever be acquired at this scale, one can see the simulation as methodology, giving experimental information from a model idealized universe, this approach has a mixture of theory and experiment and it has been labeled a “Third Kind of Science ” by computationally solving a produced physical phenomenon and can be viewed experimentally.
This dissertation describes the fracture behaviors under mode I loading (under tension deformations), the full fracture process stages under mode I, the fracture behaviors under mode I + II loading (under tension + shear deformations) and fracture with different crack length.
Chapter one contains in brief the historical scope in the fracture studies, macroscopic, microscopic view of fracture and fracture modes. In addition, this chapter reviews papers connecting with the fracture studies using experimental, finite element and atomistic computer simulations methods.
Chapter two describes the model, it was used a developed quasi-static method to study the fracture phenomenon. In brief, the crystal investigated in this work has FCC structures.
The number of atoms chains are 2400 atoms arranged in 24 rows, the Z-axis is directed along the infinite edge, the X-axis is perpendicular to the (101) plane and the Y-axis is perpendicular to (111) plane. It was chosen with the simple rare-gas solid Ar, defining the model material to perform simulation. The Lennard-Jones (LJ) (6,12) pair potential is used for interatomic interaction with a solid Ar parameters.
Periodic boundary conditions were used and it allows the atoms to move in all the directions, the model has more freedom to allow simulation of any kind of deformation and damage mechanisms. Therefore the simulation results are more reliable.
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The description of the simulation results were presented in the chapter three, during the relaxation process and at the end of it, the potential energy of the lattice per number of atoms, atomic positions displacement and lattice radial distribution function, energy contours can be monitored and plotted throughout.
From these figures, the fracture behavior in solid Ar were tested under different tension deformation levels. In addition, the crack surfaces and the crystal surfaces were studied and the results were compared with publication of atomistic computer simulation and experimental studies.
The full fracture process stages in a solid Ar were presented. The effects of mixed mode loading (I+II) were investigated. The investigation was divided
in two parts, the first part the 8» is constant and Si is variable or in the other words the applied deformation ratio is Si/Su >1 and in the second part Sj is
constant and Sji is variable or the applied deformation ratio is Si/Su <1.
The effect of different crack length, introduced in the crystal and the deformation to fracture were studied. The simulation results showed that as the crack length increased, the deformation to fracture decreased. In addition, the crack surfaces are divided in two parts, smooth part and zig-zag part.
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Chapter One
1.1 From the history of fracture study.
Galileo was seventy-two years old, his life nearly shattered by a trial for heresy before the inquisition, when he retired in 1635s to Florence to construct the “Dialogues Concerning Two New Sciences”. His first science is the study of the forces that hold objects together, and the conditions that cause fall apart, the dialogue taking place in a shipyard, by observations of craftsman building the Venation fleet. The second concerns local laws of motions governing the movement of projectiles. Although now, as in Galileo’s time, ship-builders need good answers to questions about the strength of materials, the subject has been never yielded easily to basic analysis. Galileo identified the main difficulty “one cannot reason from the small to the large, because many mechanical devices succeed on a small scale that can not exist in great size".
Nearly three hundred years Galileo wrote these lines, then the science of atomic scale began to answer the questions it had passed on the origins of strength, and the relation between large and small [1].
Robert Hooke (1635-1703) [2,3]. British physicist and instrument marker who become professor of Gresham college, London in 1660, he discovered Hooke’s law, which states that, for relatively small deformations of an object, the displacement or size of the deformation is directly proportional to the deforming force or load. At relatively large values of applied force the deformation of the elastic material is often larger then expected on the basis of Hooke’s law even through the material remains elastic and returns to its original shape and size after removing of the force.
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Hooke’s law may also be expressed in terms of stress and strain which arc usually used with fracture. Stress is the force on unit area within a material that develops as a result of the externally applied force. Strain is the relative defonnation produced by stress. So for relatively small stresses, stress is proportional to the strain.
In the 18lh century British physician and physicist Thomas Young (1773-1829) [2,3]. He described the elastic properties of a solid undergoing tension or compression in only one direction as in the case of a metal rod that after being stretched or compressed lengthwise returns to its original length, Young’s modulus is a measure of the ability of material to withstand changes in length when under lengthwise tension or compression, sometimes referred to as the modulus of elasticity Yong’s modulus is equal to the longitudinal stress divided by the strain while the ratio of the transverse strain to the longitudinal strain is called Poisson’s ratio.
Poisson, Simeon Denis (1781-1840). French mathematician is known for his work on definite integrate electromagnetic theory, probability and mechanics.
As stresses increase Young’s modulus may no longer remain constant but they decrease. Material may either flow' undergoing permanent deformation or finally fracture appears [2,3].
The history of analytical approaches to studying of fracture began in the 1920s by young English scientist Griffith A. A., he formulated his own theory of brittle fracture using elastic strain energy concepts. Griffith’s theory of brittle fracture helps us to understand why brittle fracture occurs in a material. After 30 years G. R. Irwin took Griffth’s work and applied it to ductile materials [4,5].
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1.2 Macroscopic and microscopic view of fracture.
Fracture is a very complex process that involves the nucleation and growth of dislocations, micro and macro voids, micro and macro cracks. From the point of view of mechanics, fracture , in its simplest definition can be described as a single body being separated into pieces by an imposed stress, crack initiation and propagation are essential to fracture.
From the point of view' of microscopic focus, fracture is simply the breaking of atomic bonds between atoms.
The macroscopic view of fracture is an outgrowth of theory' of linear elasticity, known as linear elastic fracture mechanics.
Linear elasticity view's all material as homogeneous elastic bodies without a microstructure. To calculate the deformations taking place when forces are applied to a body, linear elasticity uses only bulk properties of materials w'hich are well defined and easily measured, such as Young’s modulus and the Poisson ratio.
To apply linear elasticity to cracks , linear elastic fracture mechanics introduced new property of material which is known as the fracture toughness obtained by measuring the amount of force is in a carefully designed specimen before fracture.
In the microscopic view of fracture all but the smallest length scales are ignored and the crack is viewed as a disturbance in a perfect lattice of atoms.
At this length scale the forces between individual atoms are important and theories which calculate these forces, using interaction potentials. Analytically solving, the motion and displacements of atoms is difficult for large amount of atoms and instead of such calculations, it is done computationally [6].
There are only two possible types of fracture, ductile and brittle for engineering materials. In general, the main difference between brittle and ductile fracture can be attributed to the amount of plastic deformation w'hich
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the material undergoes before fracture occurs. Ductile materials demonstrate large amounts of plastic deformations before fracture while brittle materials show no plastic deformations. Macroscopically, ductile fracture surfaces have large necking regions and overall rougher appearance then a brittle fracture surface [7-11].
Attention became focused on fracture phenomenon by experimental studies and several theoretical attempts have followed by many physics and material scientists. However , there is no general mechanism of crack nucleation and growth which could be common with all the materials under different conditions. The individual ones depend on the type of bond, crystal structure, microstructure, physical conditions and geometrical sample conditions.
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1.3 Fracture modes.
For analytical purpose a crack line has been idealized as onc-dimensional defect on a flat cleavage plane. That is why, there are three modes of cracks corresponding to the different orientations of the external stress with respect to the fracture plane and they are distinguished by indices I, II and III, as it is shown in figure 1 [6].
Mode I Mode II Mode III
Fig. 1. Fracture modes.
In mode I, the stress is a tensile with principal axis normal to the cleavage plane, as it is shown in figure 1. This mode is the only one, leading to physical fracture, because, unless the external stress physically separates two surfaces on the cleavage plane, then re-welding would occur even after the stress is applied. In mode II, the stress is a shear parallel to the cut direction. In mode III, the stress is a shear, parallel to the cut in the anti-plane direction.
Mode II fracture cannot easily be observed, since slowly propagating cracks spontaneously orient themselves so as to make the mode II component
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of the loading vanish near the crack tip. Mode II fracture is however observed in cases where material is strongly anisotropic.
Both friction and earthquakes along a predefined fault are examples of mode II fracture where the binding across the fracture interface is considerably weaker then the strength of the material that comprises the bulk material.
Pure mode III fracture, although experimentally difficult to achieve, is sometimes used as a model system for theoretical study even , in this case. The equations of elasticity arc simplified considerably. Analytical solutions obtained, in this mode, have provided considerable insight to the fracture process.
A general loading situation, produced by some combination of these modes, is referred to as mixed mode fracture. However, understanding mixed mode fracture is obviously of practical importance [6,121.
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1.4 Review of experimental fracture studies.
Direct observations of crack tip deformations were made during tensile deformation by Ohr [13], using electron microscope As the stress was applied, cracks were nucleated and the mode fracture was III, as the cracks moved into the thicker sections, the mode of fracture were changed from III to I.
Using electron microscopy, Suprijead and Saka [14] also studied the dislocations formed along fracture surfaces of mode I in Si, it has been obtained that many point defects are produced during the propagation of a mode I crack in Si.
Many materials, which are used in sendee today erected in times when safety requirements in terms of notch or fracture toughness were not specified. It means, little is known about the defect tolerance in the structures under various conditions, the relation between fracture toughness temperature and loading rates [15-26].
Lorenizon and Eriksson [27] studied the influence of intermediate loading rates and temperature on the fracture toughness of ordinary Carbon-Manganese structural steels. In general this investigation showed no stable crack growth of a crack at low temperatures —30C°.
Riedle et. al. [28] performed an extensive study of the cleavage fracture of tungsten single crystal between 77 K and room temperature. The fracture surfaces clearly indicate that the intrinsic brittle fracture process is anisotropic with respect not only to the plane, but also to the direction of crack propagation.
Using scanning electron microscopic Chai et. al. [29-31], studied the intelaminar fracture toughness in mode II and mode III of laminated composites, he found that the fracture energy in mode was independent from crack extension while a rather probabilistic “resistance” behavior for mode III was exhibited which was attributed to the effect of fiber bridging.
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Mishnaevsky and Schmauder [32], developed mathematical model of damage and fracture in heterogeneous materials under dynamical loading, they derived the model from the kinetic differential equation.
The dynamics media is studied using quasistatic approximation method by Ramanathan et. al. [33].
A method to determine the time of fracture taking into account the physical mechanisms of microcrack and crack formation was developed by Mishnaevsky [34].
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