Introduction to Dislocations
272
Introduction to Dislocations
272eBook
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- Long-established academic reference by an expert author team, highly regarded for their contributions to the field.
- Uses minimal mathematics to present theory and applications in a detailed yet easy-to-read manner, making this an understandable introduction to a complex topic.
- Unlike the main competition, this new edition includes recent developments in the subject and up-to-date references to further reading and research sources.
Product Details
| ISBN-13: | 9780080966731 |
|---|---|
| Publisher: | Elsevier Science |
| Publication date: | 02/18/2011 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 272 |
| File size: | 11 MB |
| Note: | This product may take a few minutes to download. |
Read an Excerpt
Introduction to Dislocations
By D. Hull D. J. Bacon
Butterworth-Heinemann
Copyright © 2011 D. Hull and D. J. BaconAll right reserved.
ISBN: 978-0-08-096673-1
Chapter One
Defects in Crystals
1.1 CRYSTALLINE MATERIALS
Dislocations are an important class of defect in crystalline solids and so an elementary understanding of crystallinity is required before dislocations can be introduced. Metals and many important classes of non-metallic solids are crystalline, i.e. the constituent atoms are arranged in a pattern that repeats itself periodically in three dimensions. The actual arrangement of the atoms is described by the crystal structure. The crystal structures of most pure metals are relatively simple: the three most common are the body-centered cubic, face-centered cubic and close-packed hexagonal, and are described in section 1.2. In contrast, the structures of alloys and non-metallic compounds are often complex.
The arrangement of atoms in a crystal can be described with respect to a three-dimensional net formed by three sets of straight, parallel lines as in Fig. 1.1(a). The lines divide space into equal sized parallelepipeds and the points at the intersection of the lines define a space lattice. Every point of a space lattice has identical surroundings. Each parallelepiped is called a unit cell and the crystal is constructed by stacking identical unit cells face to face in perfect alignment in three dimensions. By placing a motif unit of one or more atoms at every lattice site the regular structure of a perfect crystal is obtained.
The positions of the planes, directions and point sites in a lattice are described by reference to the unit cell and the three principal axes, x, y and z (Fig. 1.1 (b)). The cell dimensions OA = a, OB = b and OC = c are the lattice parameters, and these along with the angles [??]BOC = α, [??]COA = ß and +AOB 5 completely define the size and shape of the cell. For simplicity the discussion here will be restricted to cubic and hexagonal crystal structures. In cubic crystals a = b = c and = ß = γ = 90°, and the definition of planes and directions is straightforward. In hexagonal crystals it is convenient to use a different approach, and this is described in section 1.2.
Any plane A'B'C' in Fig. 1.2 can be defined by the intercepts OA', OB' and OC' with the three principal axes. The usual notation (Miller indices) is to 1 take the reciprocals of the ratios of the intercepts to the corresponding unit cell dimensions. Thus A'B'C' is represented by
(OA/OA'. OB/OB', OC/OC') and the numbers are then reduced to the three smallest integers in these ratios.
Thus from Fig. 1.2 OA' = 2a, OB' = 3a, and OC' = 3a, the reciprocal intercepts are
(a/2a, a/3a, a/3a) and so the Miller indices of the A'B'C' plane are (322). Curved brackets are used for planes. A plane with intercepts OA, OB, and OC has Miller indices
(a/a, a/a, a/a) or, more simply, (111). Similarly, a plane DFBA in Fig. 1.3 is
(a/a, a/a, a/∞)
or (110); a plane DEGA is (a/a, a/∞, a/∞)
or (100); and a plane AB'C' in Fig. 1.2 is
(a/a, a/3a, a/3a)
or (311). In determining the indices of any plane it is most convenient to identify the plane of lattice points parallel to the plane which is closest to the origin O and intersects the principal axis close to the origin. Thus plane A'B'C' in Fig. 1.2 is parallel to ABC and it is clear that the indices are (111). Using this approach it will be seen that the planes ABC, ABE, CEA and CEB in Fig. 1.3 are (111), (11[bar.1]), (1[bar.1]1) and ([bar.1]11) respectively. The minus sign above an index indicates that the plane cuts the axis on the negative side of the origin. In a cubic crystal structure, these planes constitute a group of the same crystallographic type and are described collectively by {111}.
Any direction LM in Fig. 1.3 is described by the line parallel to LM through the origin O, in this case OE. The direction is given by the three smallest integers in the ratios of the lengths of the projections of OE resolved along the three principal axes, namely OA, OB and OC, to the corresponding lattice parameters of the unit cell. Thus, if the cubic unit cell is given by OA, OB and OC the direction LM is
[OA/OA, OB/OB. OC/OC]
or
[a/a, a/a, a/a]
or [111]. Square brackets are used for directions. The directions CG, AF, DB and EO are [11[bar.1]], [[bar.1]11], [[bar.1]1[bar.1]] and [[bar.111]] respectively and are a group of directions of the same crystallographic type described collectively by <111>. Similarly, direction CE is
[a/a, a/a, O/a]
or [110]; direction AG is
[O/a, a/a, O/a]
or [010]; and direction GH is
[a/2/a, -a/2/a, a/a]
or [[bar.11]2]. The rule that brackets and ( ) imply specific directions and planes respectively, and that < > and { } refer respectively to directions and planes of the same type, will be used throughout this text.
In cubic crystals the Miller indices of a plane are the same as the indices of the direction normal to that plane. Thus in Fig. 1.3 the indices of the plane EFBG are (010) and the indices of the direction AG which is normal to EFBG are [010]. Similarly, direction OE [111] is normal to plane CBA (111).
The coordinates of any point in a crystal relative to a chosen origin site are described by the fractional displacements of the point along the three principal axes divided by the corresponding lattice parameters of the unit cell. The center of the cell in Fig. 1.3 is 1/2, 1/2, 1/2 relative to the origin O; and the points F, E, H and I are 0, 1, 1; 1, 1, 1; 1/2, 1/2, 1; and 1, 1/2, 1 respectively.
1.2 SIMPLE CRYSTAL STRUCTURES
In this section the atoms are considered as hard spheres which vary in size from element to element. From the hard sphere model the parameters of the unit cell can be described directly in terms of the radius of the atomic sphere, r. In the diagrams illustrating the crystal structures the atoms are shown as small circles in the three-dimensional drawings and as large circles representing the full hard sphere sizes in the two-dimensional diagrams. It will be shown that crystal structures can be described as a stack of lattice planes in which the arrangement of lattice sites within each layer is identical. To see this clearly in two-dimensional figures, the atoms in one layer represented by the plane of the paper are shown as full circles, whereas those in layers above and below the first are shown as small shaded circles. The order or sequence of the atom layers in the stack, i.e. the stacking sequence, is described by labeling one layer as an A layer and all other layers with atoms in identical positions above the first as A layers also. Layers of atoms in other positions in the stack are referred to as B, C, D layers, etc.
In the simple cubic structure with one atom at each lattice site, illustrated in Fig. 1.4, the atoms are situated at the corners of the unit cell. (Note that no real crystals have such a simple atomic arrangement.) Figures 1.4(b) and (c) show the arrangements of atoms in the (100) and (110) planes respectively. The atoms touch along <001> directions and therefore the lattice parameter a is twice the atomic radius r (a = 2r). The atoms in adjacent (100) planes are in identical atomic sites when projected along the direction normal to this plane, so that the stacking sequence of (100) planes is AAA ... The atoms in adjacent (110) planes are displaced 1/2 a [square root of 2] along [110] relative to each other and the spacing of atoms along [110] is a [square root of 2]. It follows that alternate planes have atoms in the same atomic sites relative to the direction normal to (110) and the stacking sequence of (110) planes is ABABAB ... The spacing between successive (110) planes is 1/2 a [square root of 2].
In the body-centered cubic structure (bcc), which is exhibited by many metals and is shown in Fig. 1.5, the atoms are situated at the corners of the unit cell and at the centre site 1/2, 1/2, 1/2. The atoms touch along a <111> direction and this is referred to as the close-packed direction. The lattice parameter a = 4r = [square root of 3] and the spacing of atoms along <110> directions is a [square root of 2]. The stacking sequence of {100} and {110} planes is ABABAB ... (Fig. 1.5(b)). There is particular interest in the stacking of {112} type planes (see sections 6.3 and 9.7). Figure 1.6 shows two body-centered cubic cells and the positions of a set of (112) planes. From the diagrams it is seen that the stacking sequence of these planes is ABCDEFAB ..., and the spacing between the planes is a/[square root of 6].
In the face-centered cubic structure (fcc), which is also common among the metals and is shown in Fig. 1.7, the atoms are situated at the corners of the unit cell and at the centers of all the cube faces in sites of the type 0, 1/2, 1/2. The atoms touch along the <011> close-packed directions. The lattice parameter a = 4r/ [square root of 2]. The stacking sequence of {100} and {110} planes is ABABAB ..., and the stacking sequence of {111} planes is ABCABC ... The latter is of considerable importance (see Chapter 5) and is illustrated in Figs 1.7(c) and (d). The atoms in the {111} planes are in the most close-packed arrangement possible for spheres and contain three h110i close-packed directions 60° apart, as in Fig. 1.7(b).
The close-packed hexagonal structure (cph or hcp) is also common in metals. It is more complex than the cubic structures but can be described very simply with reference to the stacking sequence. The unit cell with lattice parameters a, a, c is shown in Fig. 1.8(a), together with the hexagonal cell constructed from three unit cells. There are two atoms per lattice site, i.e. at 0, 0, 0 and 2/3, 1/3, 1/2 with respect to the axes a1, a2, c. The atomic planes perpendicular to the c axis are close-packed, as in the fcc case, but the stacking sequence is now ABABAB ..., as shown in Fig. 1.8(b).
For a hard sphere model the ratio of the length of the c and a axes (axial ratio) of the hexagonal structure is 1.633. In practice, the axial ratio varies between 1.57 and 1.89 in close-packed hexagonal metals. The variations arise because the hard sphere model gives only an approximate value of the interatomic distances and requires modification depending on the electronic structure of the atoms.
If Miller indices of three numbers based on axes a1, a2, c are used to define planes and directions in the hexagonal structure, it is found that crystallo-graphically equivalent sets can have combinations of different numbers. For example, the three close-packed directions in the basal plane (001) are [100], [010] and [110]. Indexing in hexagonal crystals is therefore usually based on Miller-Bravais indices, which are referred to the four axes a1, a2, a3 and c indicated in Fig. 1.8(a). When the reciprocal intercepts of a plane on all four axes are found and reduced to the smallest integers, the indices are of the type (h, k, i, l), and the first three indices are related by
i = -(h + k) (1.1)
Equivalent planes are obtained by interchanging the position and sign of the first three indices. A number of planes in the hexagonal lattice have been given specific names. For example:
Basal plane (0001) Prism plane : first order (1[bar.1]00) ([bar.1]100); etc: Prism plane : second order (11[bar.2]0) ([bar.2]110); etc: Pyramidal plane : first order (10[bar.1]1) ([bar.1]011); etc: Pyramidal plane : second order (11[bar.2]2) ([bar.11]22); etc:
(Continues...)
Excerpted from Introduction to Dislocations by D. Hull D. J. Bacon Copyright © 2011 by D. Hull and D. J. Bacon. Excerpted by permission of Butterworth-Heinemann. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents
1. Defects in Crystals 2. Observation of Dislocations 3. Movement of Dislocations 4. Elastic Properties of Dislocations 5. Dislocations in Face-centered Cubic Metals 6. Dislocations in Other Crystal Structures 7. Jogs and the Intersection of Dislocations 8. Origin and Multiplication of Dislocations 9. Dislocation Arrays and Crystal Boundaries 10. Strength of Crystalline Solids
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