On bending of single crystal beam with continuously distributed dislocations

The theory of plastic bending of single crystal beam taking into account continuously distributed dislocations is proposed. Applying the variational-asymptotic method we reduce the energy functional of the beam to the one-dimensional energy functional which admits analytical solutions. The threshold value at the onset of plastic yielding as well as the dislocation density are found in terms of the applied bending moment. We consider also the polygonization of the bent beam after unloading and annealing and show that such state is energetically preferable. The number of polygons is estimated by comparing the surface energy of small angle tilt boundaries and the gradient terms in the bulk energy.