Continuum Mech. Thermodyn. (2010) 22:291–298 DOI 10.1007/s00161-010-0137-x ORIGINAL ARTICLE K. C. Le · Q. S. Nguyen Polygonization as low energy dislocation structure Received: 6 August 2009 / Accepted: 16 February 2010 / Published online: 5 March 2010 © Springer-Verlag 2010 Abstract Within continuum dislocation theory, one-dimensional energy functional of a bent beam, made of a single crystal, is derived. By relaxing the continuously differentiable minimizer of this energy functional, we construct a sequence of piecewise smooth deflections and piecewise constant plastic distortions reducing the energy and exhibiting polygonization. The number of polygons can be estimated by comparing the surface energy of small angle tilt boundaries with the contribution of the gradient terms from the weak minimizer in the bulk energy. Keywords Crystal · Bending · Tilt boundaries · Dislocations · Polygon 1 Introduction When a single crystal beam is plastically bent and then kept fixed in that position while annealed, one often observes its polygonization in the final stage as shown schematically in Fig. 1. Each polygon is a single crystal oriented slightly differently from its neighbors so that the boundaries between them are low angle tilt bound- aries. The dislocations align themselves into ordered arrays at these boundaries, and there are practically no dislocations inside the polygons. The mean deflection as well as the mean plastic distortion remain unchanged; however, the local curvature in each polygon becomes vanishing. The experimental observations of polygonization have been reported in the late forties of the last century (see for example [5, 6]). The first attempt of taking into account the dislocations in the plastically bent beam was made by Nye [11] who expressed the curvature of a beam caused by dislocations in terms of the dislocation density tensor bearing now his name. Read [13] and Bilby et al. [4] have extended this result to the case when the stress due to dislocations does not vanish. However, the understanding as well as the qualitative modeling of the dislocation rearrangement in the bent beam during annealing remained open, to our knowledge. This article aims at showing in the simplest case that polygonization occurs to reduce energy of the plastically bent beam which confirms the low energy dislocation structures (LEDS) hypothesis [7]. To this end we first formulate the exact two-dimensional variational problem of minimizing energy of the bent beam within the continuum dislocation theory [2, 3, 9]. Applying the variational-asymptotic procedure, we reduce the energy functional to Dedicated to the 65th birthday of V. Berdichevsky. Communicated by P. Suquet. K. C. Le (B ) Ruhr-Universität Bochum, 44780 Bochum, Germany E-mail: chau.le@rub.de Q. S. Nguyen LMS, Ecole Polytechnique, 91128 Palaiseau, France