On the analysis of pure bending of rigid-plastic beams in strain-gradient plasticity

The complete stress field, including the microstress, the moment-stress, and the line forces are derived for the pure bending of a rigid-plastic beam of rectangular cross-section in the model of strain-gradient plasticity. The workless spherical parts of the microstress and the moment-stress tensors are incorporated in the analysis. Their determination is shown to be of importance for the fulfilment of the higher-order traction boundary conditions, the physical interpretation of line forces, and their contributions to bending moments. Three equivalent methods are used to derive the moment-curvature relationship for any of the gradient-enhanced effective plastic strain measures from the considered broad class of these measures. Specific results are given for the selected choice of the stress-strain relationship describing the uniaxial tension test. Closed-form analytical expressions are obtained in the case of linear hardening, and in some cases of nonlinear hardening. The analysis of the plane-strain bending of thin foils is also presented. In this case there are two sets of line forces along the edges of the beam. The relationships between the applied bending moment and the curvature, and between the lateral bending moment and the curvature are derived and discussed. The lateral bending moment along the lateral sides of the beam, needed to keep the plane-strain mode of deformation, is one-half of the applied bending moment.