Variational principles for shakedown analysis of materials with internal rotation

Based on the extended lower bound shakedown theorem for materials with internal rotation, variational principles for elastic and plastic shakedown are presented considering the indeterminate and modified couple-stress theory, proposed by Yang et al. (Int J Solids Struct 39(10):2731–2743, 2002), combined with the first strain gradient plasticity of Fleck and Hutchinson (J Mech Phys Solids 41(12):1825–1857, 1993). Using the concepts of convex analysis, a dual principle and the optimum conditions are derived. A numerical method based on the Newton’s method is derived from the plastic shakedown variational principle to find the load amplification factor. Two illustrative numerical examples are presented: a cantilever beam subjected to a constant axial load and to a variable external bending moment and a cylindrical bar subjected to an axial load and to a variable external torque. In both cases, the external torque and external bending moment vary arbitrarily between prescribed symmetric bounds. The results are compared with those obtained considering classical plastic shakedown theory in order to see the influence of the length scale on the load amplification factor.