A dislocation experiences an opposing force (the basic level of lattice friction) when it tries to move through an otherwise perfect crystal (i. e., without any other defect acting as obstacles). The corresponding stress needed to move a dislocation in a particular direction in the crystal is known as Peierls-Nabarro (P-N) stress. The P-N stress is a direct consequence of the periodic force field present in crystal lattice and is very sensitive to any changes in the individual atom positions. That is, it is a function of the dislocation core structure, and hence developing a single analytical expression is difficult. However, the analysis forwarded by Peierls (1940) and Nabarro (1947) still gives us some important qualitative understanding that is of definite value. The P-N stress (tp_N) is given by the following relation:

where w is the dislocation core width, b is the distance between atoms in the slip direction, that is, the Burgers vector of the dislocation involved, G is the shear mod­ulus, and v is the Poisson’s ratio of the material. It can be shown that for screw dislocations, w is close to the interplanar spacing between slip planes (d); whereas

for edge dislocations, w is given by (d/(1 — n)). Dislocation core widths generally seem to vary between b and 5b (sometimes on the order of 10b for ductile metals), depending again on the interatomic potential and crystal structure. The related energy barrier is called Peierls energy.

Despite its limitations, the concept of P-N stress explains some important quali­tative aspects of plastic deformation in various crystalline materials. From Eq. (4.2), one can see that a wider dislocation core (i. e., with larger w) leads to lower value of P-N stress. This situation arises in ductile metals where the dislocation core is highly distorted and is not localized. However, ceramic materials (with covalent and/or ionic bonds) that show a very low or no ductility at lower temperatures have dislocations of narrower width. This also means that the P-N stress in ceramic materials is correspondingly high. Also, the presence of electrostatic forces in ceramic materials makes the dislocation movement difficult leading to lower plas­ticity. However, some ductility in ceramic materials can be obtained by increasing temperature that provides for thermal activation surmounting the relevant Peierls barrier. The relative higher P-N stress in BCC metals compared to FCC metals can also be explained using Eq. (4.2). In FCC metals, slip occurs on the close-packed planes that are greater distance apart, that is, d is higher). In the close-packed struc­ture, the magnitude of the Burgers vector (b) is smaller. That means d > b. It makes the magnitude of tP—N smaller. The opposite thing happens in BCC crystals that contain loosely packed slip planes with close separation from each other, that is, d < b. Again, from Eq. (4.2), we can see that tP—N will be higher in BCC and like typical not-so close-packed crystal lattices.

4.1.3