A long copper crystal ( , ) is given a dislocation along the axis by Volterra's…

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Condensed Matter, Materials & Soft Matter · Crystal structure, bonding and diffraction

Bending, Ringing, Breaking and Flowing, Ch Dislocations and the onset of plastic flow

Problem

A long copper crystal (

G = 48\,\mathrm{GPa}

\nu = 0.34

) is given a dislocation along the

z

axis by Volterra's construction. It is cut on the half-plane

y=0

x<0

, the face above the cut is slid over the face below by

0.222\,\mathrm{nm}

along

\hat{\mathbf x}

and

0.128\,\mathrm{nm}

along

\hat{\mathbf z}

, and the faces are reglued. ((a)) From the circuit integral

\oint_C\mathrm{d}\mathbf u

, identify the Burgers vector, its magnitude and its angle to the line. Is it the copper lattice translation

b=0.256\,\mathrm{nm}

? ((b)) Split

\mathbf b

into a screw part along the line and an edge part across it. Show that the sum of a screw field with the first and the quoted edge field with the second returns

\mathbf b

on every circuit round the line. ((c)) Show that the strain energy density of the summed field has no cross term between the two parts, so that

U/L=(G/4\pi)\left[b_s^2+b_e^2/(1-\nu)\right]\ln(R/r_0)

. ((d)) With

R=1\,\mathrm{\mu m}

and

r_0=b

, find

U/L

and the energy per atomic length

b

of line in electronvolts.

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