Geometric theory of defects in solids
Geometric theory of defects in solids
Geometric theory of defects in solids
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y<br />
Wedge dislocation <strong>in</strong> elasticity <strong>theory</strong><br />
R<br />
z<br />
−2πθ<br />
x<br />
r, ϕ,<br />
z<br />
ui<br />
- cyl<strong>in</strong>drical coord<strong>in</strong>ates<br />
{ u(),() r v r ϕ,0}<br />
= - displacement vector<br />
Boundary conditions:<br />
u | = 0, u | = 0, u | =−2 r, ∂ u | = 0<br />
r r= 0 ϕ ϕ= 0 ϕ ϕ= 2π πθ r r r=<br />
R<br />
vr ()<br />
u<br />
∂r( r∂ru)<br />
− = D - elasticity equations<br />
r<br />
1−<br />
2σ D =− θ , σ - Poisson ratio<br />
1 − σ<br />
D<br />
c2<br />
u = rln r+ cr 1 + , c1,2<br />
= const - a general solution<br />
2<br />
r<br />
2 ⎛ 1−2σ<br />
r ⎞ 2 2⎛ 1−2σ<br />
r 1 ⎞ 2<br />
dl = ⎜1+ θ ln ⎟dr + r ⎜1+ θ ln + θ ⎟dϕ<br />
⎝ 1−σ R⎠ ⎝ 1−σ R 1−σ<br />
⎠<br />
<strong>in</strong>duced metric θ 1, r ∼ R<br />
= −θ r<br />
θ - deficit angle<br />
11
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