Metals- Peierls-Nabarro Stress and Dislocation Mechanics Flashcards
What is the Peierls-Nabarro stress?
The minimum shear stress to move a dislocation
Why is the Peierls-Nabarro stress needed?
As a dislocation moves bonds need to be broken and reformed l an energy barrier exists to move the dislocation from one position to the next. A shear stress is therefore required to overcome this energy barrier
Why is the Peierls-Nabarro stress difficult to calculate?
Because of the directionality of the interatomic bonds
Formula for Peierls-Nabarro stress
τp=(2G/(1-ν))exp(-2πw/b) Where G is the shear modulus ν is Poisson’s ratio b is the Burgers vector w is the dislocation width
What is the width of a dislocation?
The distance over which atoms are significantly dislocated from their perfect crystal positions. Can be taken to be the region in which the atoms are displaced by over b/4
How much are atoms displaced by at the core of a dislocation?
b/2
When is Peierls stress minimised?
For slip on the close packed planes in the close packed directions for which the atomic slip distance is a minimum
Why do edge dislocations tend to be more mobile than screw dislocations?
A planar core gives a lower value of τsubp compared to a screw dislocation core.
What do wider dislocations mean?
Lower Peierls-Nabarro stress
Temperature dependence of PN stress
For FCC crystals essentially no dependence. For BCC crystals PN stress decreases significantly with increasing temperature. This is the origin of the DBTT
Peierls stress and dislocation widths for different material classes
FCC metals: w is wide, τp very small
BCC metals: w is narrow, τp moderate
Ionic solids (salts): w is narrow, τp moderate
Covalent solids (ceramics): w is very narrow, τp large
Deriving the force acting on a dislocation formula
Edge dislocation length L experiences a force per unit length F. So total force acting over the dislocation is FL. Work done is force times distance so work to move dislocation through distance d is FLd. This must be balanced against work done by shear stress to shear crystal by distance d which is τb. This acts over the entire slip plane area so equal to τLdb. Equate work done FLd=τLdb. Simplify to
F=τb
What is the overall strain energy of a dislocation equal to?
The sum of the strain energy associated with the elastic distortion of the crystal around a dislocation and the strain energy of the dislocation core. Can only calculate the energy associated with the elastic distortion of the crystal around a dislocation
Elastic strain energy of a dislocation formula
Frank’s rule
Λ=aGb^2= roughly 1/2 Gb^2
This is per unit length
Where does line tension come from?
Consider the line tension in a dislocation caused by extending its length (e.g stretching into curve). Tension in the line analogous to surface tension in bubble. Tension T produced because as the dislocation increases in length it’s energy increases resulting in a restoring force equal to the increase in energy per unit length, which must be opposed by the tension in the dislocation.
