HOMEWORK 4 SOLUTIONS

Exercise 4.6

Exercise 4.9

a)

It is vectorially incorrect

b) Dislocation energy is Gb²/2

It is energetically favorable

Exercise 4.10

a) Assuming array a 2-D of dislocations

The average space between dislocations lines in the:

Annealed condition, L=3.16 mm

Deformed condition L=0.316 nm

Assuming array a 3-D of dislocations

Annealed condition, R=2.72 mm

Deformed condition R=0.272 nm

b) Line energy

Using

Edge dislocation

Screw dislocation

In the annealed condition U_s=3.14 X10^-9 J/m and U_e=4.5 X10^-9 J/m. In the deformed condition U_s=1.29 X10^-9 J/m. and U_e=1.85 X10^-9 J/m

Exercise 4.24

The dislocation line tension T=(1/2)Gb²

The force on a dislocation per unit length is given by: F=tb, where t is the shear stress and b is the burgers vector, and F=T/r, T being the line tension and r is the radius to which the dislocation is bent.

The shear stress necessary to bow the dislocation into a semicircle of radius , r is t=F/b=T/br

Exercise 4.25at the grain boundary

The stress acting on the source of dislocations due to the presence of n dislocations and due to the applied stress is:

The length of the pileup:

So,

Therefore,