Elasticity_ Barber

202 13 Forces dislocations and cracks
13.2 Dislocations
The term dislocation has related, but slightly different meanings in Elasticity
and in Materials Science. In Elasticity, the material is assumed to be a continuum
— i.e. to be infinitely divisible. Suppose we take an infinite continuous
body and make a cut along the half-plane x > 0, y = 0. We next apply equal
and opposite tractions to the two surfaces of the cut such as to open up a
gap of constant thickness, as illustrated in Figure 13.1. We then slip a thin
slice of the same material into the space to keep the surfaces apart and weld
the system up, leaving a new continuous body which will now be in a state of
residual stress.
Figure 13.1: The climb dislocation solution.
The resulting stress field is referred to as the climb dislocation solution. It can
be obtained from the stress function of equation (13.1) by requiring that there
be no net force at the origin, with the result C 1 =0 (see equation 13.10). We
therefore have
φ = C 3 r ln(r) cos θ . (13.15)
The strength of the dislocation can be defined in terms of the discontinuity
in the displacement u θ on θ =0, 2π, which is also the thickness of the slice of
extra material which must be inserted to restore continuity of material. This
thickness is
δ = u θ (0) − u θ (2π) = − π(κ + 1)C 3
. (13.16)
2µ
Thus, we can define a climb dislocation of strength B y as one which opens a
gap δ =B y , corresponding to
where
C 3 = −
2µB y
π(κ + 1) , (13.17)