Lower bound shakedown theorem for materials with internal rotation

• No shakedown theory is available for the analysis of materials with internal rotation.
• A lower bound shakedown theorem including length scale effects is presented.
• The proposed theorem introduces the new concepts of mechanics of generalized continua.
• An upper bound shakedown theorem may be obtained through a dualization procedure.

Materials with internal rotation belong to the class of polar materials, and they have a mechanical description suitable to model the behavior of microdevices, like MEMS, and materials such as grains, cellular solids and fiber-reinforced polymers. These types of materials have length scales that need a specific elastoplastic description. The shakedown theory is a useful tool for the analysis of mechanical components that are subjected to variable loads during some interval of time and for which only the limits of load variation are known. Although this may be a common mechanical situation, no shakedown theory is available for the analysis of materials with internal rotation. In this paper, a lower bound shakedown theorem for the analysis of materials with internal rotation is presented. The theorem is an extension of Melan's classical lower bound theorem and uses a linear elastic couple-stress theory combined with the first strain gradient plasticity of Fleck and Hutchinson. Considering a self-equilibrated couple-stress field, it is shown that the plastic dissipation is bounded and that the material shakes down for any given combinations of loads and moments. As a first attempt for a shakedown theory for this type of material, this theorem can give rise to variational formulations useful for numerical computations, similar to those previously used for the classical shakedown theory.