Section 13.4 modelled the restoring shear stress between two sliding close-packed…

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

Condensed Matter Physics, Ch Real crystals: vacancies, dislocations and the strength of metals

Problem

Section 13.4 modelled the restoring shear stress between two sliding close-packed planes as a sine wave, matched to Hooke's law at small displacement. Suppose instead the restoring stress rises linearly from

\tau_{\rm s}=0

x=0

to a peak

\tau_{\rm th}

x=b_{\rm B}/4

, then falls linearly back to zero at

x=b_{\rm B}/2

, with the same shape repeating over each half period. ((a)) Find the initial slope

\mathrm{d}\tau_{\rm s}/\mathrm{d}x

near

x=0

in terms of

\tau_{\rm th}

and

b_{\rm B}

. ((b)) Match that slope to Hooke's law,

\tau_{\rm s} = Gx/d_{\rm s}

, to express

\tau_{\rm th}

in terms of

G

b_{\rm B}

and

d_{\rm s}

. ((c)) Evaluate

\tau_{\rm th}

for copper (

G=48\ \mathrm{GPa}

), taking

d_{\rm s} \approx b_{\rm B}

, and compare it with the sine-based estimate of

7.6\ \mathrm{GPa}

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