Chapter-4-Part-II - OUTLINE 4.3 Line Defects 4.3.1...

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Unformatted text preview: OUTLINE 4.3 Line Defects 4.3.1 Dislocations 3) Theoretical Shear Stress b) Edge Dislocations c) Burgers Circuit and Burgers Vector d) Screw Dislocations e) Mixed Dislocations i) Opposite sign Dislocations g) Dislocation Density 4.4 Interfacial Defects 4.4.1 Free Surface 4.4.2 Grain Boundaries 4.4.3 Stacking Faults 4.4.5 Interphase Boundaries 4.5 Bulk Defects 4.3 Line Defects 4.3.1 Dislocations a) Theoretical Shear Stress Now we are going to talk about another class of defects called linear defects. These defects, called dislocations, are the main mechanism operating when a material is deformed plastically. Currently, several techniques are available for the direct observation of dislocations. The Transmission Electron Microscope is probably the most utilized in this respect. However, the discovery of dislocations was not an easy task. Let me illustrate the challenge involved in the discovery of these defects. Let us consider two lattice planes, as depicted in Figure 102. We apply a shear stress to this material, so that the top layer of atoms is required to move across the bottom row of atoms. When the atoms move, they will go up and down the lattice potential and thus, this movement can be represented by a sinusoidal relationship in the form . 272x r=rm Sin—6- where x is the shear translation of the two rows away from the equilibrium position, b is the interplannar spacing and r is the shear stress. Let us now assume that the strains involved are small. Thus, we can write 81 sin@7%)s “0%, thus 271x m T T=T We also know that Assuming b s a => rm E % 00.00 a0T0000 T Figure 102: The process of slip in a perfect crystal. Now, the problem with this model is that if an experiment is performed, the observed result is 10mm“ =10" —10'8u. In other words, crystals deform plastically at stresses X0000 of its theoretical strength. This is puzzling! What has gone wrong? Well, the discrepancy between the computed and the real yield stresses is because real crystals contain defects called dislocations, which are much easier to move than shearing a row of atoms across a bottom row of atoms. 82 b) Edge Dislocation Let us start the discussion about dislocations by looking at the simplest case, the edge dislocation. Imagine the following sequence of events: 1) take a perfect crystal, 2) make a cut in the crystal, 3) open the cut and 4) insert an extra plane of atoms (Figure 103). The end result is an edge dislocation. The way we usually depict dislocations is by the symbol .L. (a) (b) dislocation (c) Figure 103: Sequence of events to create an edge dislocation. What does characterize an edge dislocation? In fact, there are two important parameters, namely a) line direction and b) Burgers vector. The line direction is represented by a vector § (segment CD in Figure 104). The Burgers vector is determined by constructing a Burgers circuit around the dislocation. This can be illustrated by looking into the plane perpendicular to the dislocation line (Figure 105). Figure 104: Three-dimensional representation of an edge dislocation. c) Burgers vector and Burgers circuit We start drawing a circuit around the dislocation, starting at point S and going in the clockwise direction (Figure 105a). Then we count how many lattice points we touch until the circuit is closed. Then we apply the same procedure around a perfect crystal (Figure 83 lOSb). We realize that there is a vector required to complete the circuit. This is the Burgers vector b. In other words, the Burgers vector is the displacement caused by the presence of the edge dislocation. In the case of an edge dislocation we note that the Burgers vector b is normal to the line direction g. Figure 105: Burgers circuit in a crystal containing a dislocation, (a) and in a perfect crystal, (b). (1) Screw Dislocation On other type of dislocations is the screw dislocation. This occurs when a unit of a growing crystal does not join exactly next to its neighbor but grows at an angle into a slightly upper plane (Figure 106). As shown in Figure 106, the screw dislocation has its Burgers vector parallel to the dislocation line direction. Figure 106: Three-dimensioual representation of a screw dislocation with a Burgers circuit drawn. The Burgers vector is parallel to the dislocation line direction. 84 e) Mixed Dislocation In the real world, the dislocation line lies at an arbitrary angle to its Burger vector, and thus the dislocation is neither, a pure edge or a pure screw dislocation. In fact, it has a mixed character composed by edge and screw components (Figure 107). As an example let us look into the dislocation configuration shown in Figure 108. Where is the character of the dislocation screw and where is edge? The way is to find out where the dislocation line direction is parallel or perpendicular to the Burgers vector. bedge b b SCI‘CW Figure 107: Edge and screw component of an edge dislocation. screvv edge ———————-> b Figure 108: Mixed dislocation loop exhibiting regions of edge and screw character. t) Opposite Sign Dislocations We saw before that when drawing the Burgers circuit we were following the clockwise direction. What would happen if we had chosen the counter-clockwise direction? We see that reversing the line sense reverses the direction of the Burgers vector for a particular dislocation. So, a positive dislocation (.L) is a right-handed screw dislocation, whereas a negative dislocation is a left-handed screw dislocation. What happens when a positive 85 and a negative dislocation come together in the same plane? A phenomenon called annihilation occurs (Figure 109). As a result, the Burgers vectors cancel each other, which means that the perfect lattice is restored. Before After Annihilation Annihilation L- ........ V. T Figure 109: Two edge dislocations of opposite sign come together on the same plane. This mechanism restores the perfect lattice because the Burgers vectors of each dislocation cancel each other. g) Dislocation Density One of the important parameters related to dislocations is the knowledge of the amount of dislocation length per unit volume. This is called the dislocation density and is given by p = total length of dislocation/unit volume (cm'z) Typically, the density of dislocations is as follows 10‘ — 10‘ /cmz when annealed 1011-10” 1 when deformed cm 4.4 Interracial Defects 4.4.1 Free surfaces All solid materials have finite sizes. As a result, the atomic arrangement at the surface is different from within the bulk. Typically, surface atoms form the same crystal structure as in the bulk but the unit cell have slightly larger lattice parameter. In addition, surface atoms will have more freedom to move and thus, higher entropy. As shown in Figure 110, the surface atoms are not bonded in the direction normal to the surface plane. Hence, if the energy of each bond is 8/2 , then for each surface atom not bonded, there is an excess energy of This excess in energy can be determined from the expression Gs= Es — T85, 86 where G5 is the surface free energy. For a pure metal, the term E8 can be estimated from the latent heat of melting + the latent heat of vaporization. Moreover, the surface atoms will have extra entropy, which will counteract the increase in energy. We may now understand that different crystal surfaces should have different values for E5, depending on the number of broken bonds. As an example, let us depict the atomic stacking on the (111) planes of a FCC material (Figure 111). We note that there are 3 atoms on the second layer sitting on the first layer. This means that, if we remove the second layer, the atoms on the bottom layer will have some bonds missing. Applying this same concept to the (200) plane of a FCC material, we note that now there will be four atoms sitting on top of the first layer (Figure 111) and thus more bonds will be absent, if the material terminates at the first layer. Figure 111: A different surface atomic arrangement will lead to different surface energies. 87 4.4.2 Grain boundaries Grain boundaries separate regions of different crystallographic orientation, i.e., separate the various single crystals. The simplest form of a grain boundary is called a tilt boundary because the mison'entation is in the form of a simple tilt about an axis (Figure 112). Atomistically, the grain boundary can be depicted as composed of a parallel array of edge dislocations (Figure 113). The niisorientation angle 6 of a tilt boundary is given by b . . = B where b 15 the Burgers vector and D is the distance between dislocations. This relationship can be better shown by looking at Figure l 13. / Figure 112: Tilt Boundary Figure 113: Low angle tilt boundary modeled by an array of edge dislocations. Another type of grain boundary is the twist boundary (Figure 114). In this case, the angle of misorientation is parallel to the boundary. Twist boundaries can be represented by an array of screw dislocations. Both tilt and twist boundaries are low angle boundaries, i.e., the angle of misorientation is less than 10 degrees. What happens when a grain boundary has a misorientation greater than 10 degrees? In this case, it is called a high- angle boundary and can no longer be represented by an array of dislocations because the spacing between the dislocations would be so small that they would lose their individual identity. 88 Twin 11 I Figure 114: Twist Boundary We know that grain boundaries have also an interfacial energy, due to the disruption of the atomic periodicity. However, will they have higher or lower energies than the free surfaces? The answer is lower energies. This is because grain boundaries exhibit some distortion in the type of bonds they form but there are no absent bonds. Some boundaries, which have significant lower energies than other high-angle boundaries are called Special high-angle boundaries. These boundaries occur at particular misorientations and boundary planes that allow the two adjoining lattices to fit together with relatively little distortion. One example is the twin boundary (Figure 115). . - . .. o - o -= o : I Molvil . .. . o '- o = ' "' (Illl‘l’winninq Plene (m) Twinning ‘ _ _, 5 Plane 0 - I - -' o - o w 0 ~ 0 - - .~ o v o .- o o o Maui: . K - g o o I . o a Figure 115: Twin Boundary. The periodicity across the boundary is maintained and thus the boundary has low energies. Now, let us discuss some practical consequences of grain boundaries. As you probably know, most of the materials utilized in our daily lives are polycrystalline materials. This means that the materials are composed by many crystals, separated by grain boundaries (Figure 116). Let us now perform an experiment and heat the material. What happens to the grain boundaries? In fact the grain boundaries will migrate and grain growth will occur, where the small crystals will disappear at the expense of the larger ones. In addition, if there are impurities in the material, they will tend to go to the grain boundaries to minimize the overall energy of the system. 89 Single-Cwstal Single-Crystal Single-Cwstal Grain Boundaries Figure 116: Set of three grains separated by grain boundaries. 4.4.4 Stacking Faults. The closed-packed FCC and HCP structures can be generated by stacking closed-packed layers on top of one another, in the fashion illustrated in Figure 117. In the case of an FCC structure, given a layer A, close packing can be extended by stacking the next layer so that its atoms occupy B or C sites. Here, A, B, and C refer to the three possible layer positions in a projection normal to the (111) plane of the FCC structure. A closed packed is generated, provided that no two layers of the same letter index, such as AA are stacked in juxtaposition to one another. The sequence corresponding to an FCC crystal is ABCABCABC... On the other hand, the stacking sequence for HCP on the (0001) plane is ABABAB...In the past we have discussed that a small shift in the position of the atoms, can transform the FCC to HCP and vice-versa. Thus, when alayer of atoms is removed from the normal stacking FCC sequence, one finds ABCABCABCABC lulu t ABCABCA where at some point, the C layer becomes a A layer, the A layer becomes a B layer, and the B layer becomes the C layer. As a result we get the stacking sequence ABABABCABCA This is a stacking fault because the FCC stacking is now changed to a HCP stacking. 90 ABCABC Figure 117: Stacking sequence in the closed-packed planes of HCP (left figure) and FCC (right figure) structures. 4.4.5 biterphase boundaries Before talking about interphases, let me introduce first the concept of a phase. A phase is a portion of the material that has a) a particular crystal structure and b) a certain chemical composition. Thus, the interphase boundaries are the boundaries, which separate these regions. An example of this is the dentist’s dn'll, which is a mixture of small crystals of tungsten carbide surrounded by a matrix of cobalt. The interfaces between the tungsten carbide and the cobalt matrix are interphase boundaries. 4.5 Bulk defects These volume defects are most of the times introduced during the production of materials. In some cases, impurities in the material combine with each other and fonn inclusions. These are second-phase particles, which may affect considerably the mechanical properties of materials. One other type of defects are the casting defects. These can be cavities and gas holes produced under certain conditions of temperature and pressure. A different type of volume defects are the cracks formed during heating and cooling cycles, or during formability processes. Finally, welding defects can be formed during the welding procedure, due to the fact that the heat generated during welding is not uniform. As a consequence, a region affected by the heat is produced, where the properties change gradually away fi'om the heat source. 91 ...
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This note was uploaded on 09/04/2011 for the course ME 311 taught by Professor Meyers during the Spring '08 term at University of Texas at Austin.

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Chapter-4-Part-II - OUTLINE 4.3 Line Defects 4.3.1...

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