BENDING PRODUCED BY DISLOCATIONS IN A BEAM

There is a correspondence between the deformations predicted by the theory of ideal fibre-reinforced materials (IFRM) and deformations produced by dislocations. This was investigated in the present work. Slip on central sections of a beam was modelled using continuous distributions of infinitesimal edge dislocations. It is shown that the governing integral equation in this problem is very similar to that arising in the Bilby, Cottrell and Swinden (BCS) model of a crack with plastic zones at the tip. In the BCS model, the accumulation of dislocations near the tips of the crack controls the crack tip opening displacement. In the present problem, a similar accumulation together with the presence of free surfaces requires that the beam bend plastically in the regions of the loading points consistent with the form predicted by IFRM theory. Numerical solutions were obtained for the spread of plasticity with increasing shear stress and for the plastic curvatures induced by yielding. The results are interpreted as representing the early stages in the deformation of a beam as its behaviour changes from purely elastic towards the IFRM limit.