Bending, Ringing, Breaking and Flowing, Ch Dislocations and the onset of plastic flow
Problem
A straight screw dislocation, Burgers vector bz^ and line along z^, lies at depth d below the flat free surface of a large copper crystal (G=48GPa, b=0.256nm). Put the surface on the plane y=0, the crystal in y>0 and the line at (0,d). (a) Write the screw field σθz=Gb/2πr about the line in Cartesian components σxz and σyz, and show that on its own it puts a shear traction σyz(x,0)=Gbx/2π(x2+d2) on the surface, which the free surface cannot carry. (b) Add the field of a second screw, Burgers vector −bz^, placed at the mirror point (0,−d) outside the crystal. Show that the sum satisfies the Navier equation everywhere in the crystal except on the real line and leaves the surface free of traction. (c) Use the Peach-Koehler force to find the force per unit length on the real dislocation, with its direction, and evaluate it for d=50nm. (d) Find the resolved shear stress that would have to be applied to hold the dislocation at that depth.